Fields serve as foundational notions in several mathematical domains. This includes different branches ofmathematical analysis, which are based on fields with additional structure. Basic theorems in analysis hinge on the structural properties of the field of real numbers. Most importantly for algebraic purposes, any field may be used as thescalars for avector space, which is the standard general context forlinear algebra.Number fields, the siblings of the field of rational numbers, are studied in depth innumber theory.Function fields can help describe properties of geometric objects.
Informally, a field is a set, along with twooperations defined on that set: an addition operationa +b and a multiplication operationa ⋅b, both of which behave similarly as they do forrational numbers andreal numbers. This includes the existence of anadditive inverse−a for all elementsa and of amultiplicative inverseb−1 for every nonzero elementb. This allows the definition of the so-calledinverse operations, subtractiona −b and divisiona /b, asa −b =a + (−b) anda /b =a ⋅b−1.Often the producta ⋅b is represented by juxtaposition, asab.
Formally, a field is asetF together with twobinary operations onF calledaddition andmultiplication.[1] A binary operation onF is a mappingF ×F →F, that is, a correspondence that associates with each ordered pair of elements ofF a uniquely determined element ofF.[2][3] The result of the addition ofa andb is called the sum ofa andb, and is denoteda +b. Similarly, the result of the multiplication ofa andb is called the product ofa andb, and is denoteda ⋅b. These operations are required to satisfy the following properties, referred to asfield axioms.
These axioms are required to hold for allelementsa,b,c of the fieldF:
Associativity of addition and multiplication:a + (b +c) = (a +b) +c, anda ⋅ (b ⋅c) = (a ⋅b) ⋅c.
Commutativity of addition and multiplication:a +b =b +a, anda ⋅b =b ⋅a.
Additive inverses: for everya inF, there exists an element inF, denoted−a, called theadditive inverse ofa, such thata + (−a) = 0.
Multiplicative inverses: for everya ≠ 0 inF, there exists an element inF, denoted bya−1 or1/a, called themultiplicative inverse ofa, such thata ⋅a−1 = 1.
Distributivity of multiplication over addition:a ⋅ (b +c) = (a ⋅b) + (a ⋅c).
An equivalent, and more succinct, definition is: a field has two commutative operations, called addition and multiplication; it is agroup under addition with0 as the additive identity; the nonzero elements form a group under multiplication with1 as the multiplicative identity; and multiplication distributes over addition.
Even more succinctly: a field is acommutative ring where0 ≠ 1 and all nonzero elements areinvertible under multiplication.
Fields can also be defined in different, but equivalent ways. One can alternatively define a field by four binary operations (addition, subtraction, multiplication, and division) and their required properties.Division by zero is, by definition, excluded.[4] In order to avoidexistential quantifiers, fields can be defined by two binary operations (addition and multiplication), two unary operations (yielding the additive and multiplicative inverses respectively), and twonullary operations (the constants0 and1). These operations are then subject to the conditions above. Avoiding existential quantifiers is important inconstructive mathematics andcomputing.[5] One may equivalently define a field by the same two binary operations, one unary operation (the multiplicative inverse), and two (not necessarily distinct) constants1 and−1, since0 = 1 + (−1) and−a = (−1)a.[a]
Rational numbers have been widely used a long time before the elaboration of the concept of field.They are numbers that can be written asfractionsa/b, wherea andb areintegers, andb ≠ 0. The additive inverse of such a fraction is−a/b, and the multiplicative inverse (provided thata ≠ 0) isb/a, which can be seen as follows:
The abstractly required field axioms reduce to standard properties of rational numbers. For example, the law of distributivity can be proven as follows:[6]
Thereal numbersR, with the usual operations of addition and multiplication, also form a field. Thecomplex numbersC consist of expressions
a +bi, witha,b real,
wherei is theimaginary unit, i.e., a (non-real) number satisfyingi2 = −1.Addition and multiplication of real numbers are defined in such a way that expressions of this type satisfy all field axioms and thus hold forC. For example, the distributive law enforces
It is immediate that this is again an expression of the above type, and so the complex numbers form a field. Complex numbers can be geometrically represented as points in theplane, withCartesian coordinates given by the real numbers of their describing expression, or as the arrows from the origin to these points, specified by their length and an angle enclosed with some distinct direction. Addition then corresponds to combining the arrows to the intuitive parallelogram (adding the Cartesian coordinates), and the multiplication is – less intuitively – combining rotating and scaling of the arrows (adding the angles and multiplying the lengths). The fields of real and complex numbers are used throughout mathematics, physics, engineering, statistics, and many other scientific disciplines.
In antiquity, several geometric problems concerned the (in)feasibility of constructing certain numbers withcompass and straightedge. For example, it was unknown to the Greeks that it is, in general, impossible to trisect a given angle in this way. These problems can be settled using the field ofconstructible numbers.[7] Real constructible numbers are, by definition, lengths of line segments that can be constructed from the points 0 and 1 in finitely many steps using onlycompass andstraightedge. These numbers, endowed with the field operations of real numbers, restricted to the constructible numbers, form a field, which properly includes the fieldQ of rational numbers. The illustration shows the construction ofsquare roots of constructible numbers, not necessarily contained withinQ. Using the labeling in the illustration, construct the segmentsAB,BD, and asemicircle overAD (center at themidpointC), which intersects theperpendicular line throughB in a pointF, at a distance of exactly fromB whenBD has length one.
Not all real numbers are constructible. It can be shown that is not a constructible number, which implies that it is impossible to construct with compass and straightedge the length of the side of acube with volume 2, another problem posed by the ancient Greeks.
In addition to familiar number systems such as the rationals, there are other, less immediate examples of fields. The following example is a field consisting of four elements calledO,I,A, andB. The notation is chosen such thatO plays the role of the additive identity element (denoted 0 in the axioms above), andI is the multiplicative identity (denoted1 in the axioms above). The field axioms can be verified by using some more field theory, or by direct computation. For example,
A ⋅ (B +A) =A ⋅I =A, which equalsA ⋅B +A ⋅A =I +B =A, as required by the distributivity.
This field is called afinite field orGalois field with four elements, and is denotedF4 orGF(4).[8] Thesubset consisting ofO andI (highlighted in red in the tables at the right) is also a field, known as thebinary fieldF2 orGF(2).
The axioms of a fieldF imply that it is anabelian group under addition. This group is called theadditive group of the field, and is sometimes denoted by(F, +) when denoting it simply asF could be confusing.
Similarly, thenonzero elements ofF form an abelian group under multiplication, called themultiplicative group, and denoted by or just, orF×.
A field may thus be defined as setF equipped with two operations denoted as an addition and a multiplication such thatF is an abelian group under addition, is an abelian group under multiplication (where 0 is the identity element of the addition), and multiplication isdistributive over addition.[b] Some elementary statements about fields can therefore be obtained by applying general facts ofgroups. For example, the additive and multiplicative inverses−a anda−1 are uniquely determined bya.
The requirement1 ≠ 0 is imposed by convention to exclude thetrivial ring, which consists of a single element; this guides any choice of the axioms that define fields.
In addition to the multiplication of two elements ofF, it is possible to define the productn ⋅a of an arbitrary elementa ofF by a positiveintegern to be then-fold sum
a +a + ... +a (which is an element ofF.)
If there is no positive integer such that
n ⋅ 1 = 0,
thenF is said to havecharacteristic0.[11] For example, the field of rational numbersQ has characteristic 0 since no positive integern is zero. Otherwise, if thereis a positive integern satisfying this equation, the smallest such positive integer can be shown to be aprime number. It is usually denoted byp and the field is said to have characteristicp then.For example, the fieldF4 has characteristic2 since (in the notation of the above addition table)I +I = O.
IfF has characteristicp, thenp ⋅a = 0 for alla inF. This implies that
(a +b)p =ap +bp,
since all otherbinomial coefficients appearing in thebinomial formula are divisible byp. Here,ap :=a ⋅a ⋅ ⋯ ⋅a (p factors) is thepth power, i.e., thep-fold product of the elementa. Therefore, theFrobenius map
F →F :x ↦xp
is compatible with the addition inF (and also with the multiplication), and is therefore a field homomorphism.[12] The existence of this homomorphism makes fields in characteristicp quite different from fields of characteristic0.
AsubfieldE of a fieldF is a subset ofF that is a field with respect to the field operations ofF. EquivalentlyE is a subset ofF that contains1, and is closed under addition, multiplication, additive inverse and multiplicative inverse of a nonzero element. This means that1 ∊E, that for alla,b ∊E botha +b anda ⋅b are inE, and that for alla ≠ 0 inE, both−a and1/a are inE.
Field homomorphisms are mapsφ:E →F between two fields such thatφ(e1 +e2) =φ(e1) +φ(e2),φ(e1e2) =φ(e1) φ(e2), andφ(1E) = 1F, wheree1 ande2 are arbitrary elements ofE. All field homomorphisms areinjective.[13] Ifφ is alsosurjective, it is called anisomorphism (or the fieldsE andF are called isomorphic).
A field is called aprime field if it has no proper (i.e., strictly smaller) subfields. Any fieldF contains a prime field. If thecharacteristic ofF isp (a prime number), the prime field is isomorphic to the finite fieldFp introduced below. Otherwise the prime field is isomorphic toQ.[14]
Finite fields (also calledGalois fields) are fields with finitely many elements, whose number is also referred to as the order of the field. The above introductory exampleF4 is a field with four elements. Its subfieldF2 is the smallest field, because by definition a field has at least two distinct elements,0 and1.
In modular arithmetic modulo 12,9 + 4 = 1 since9 + 4 = 13 inZ, which divided by12 leaves remainder 1. However,Z/12Z is not a field because12 is not a prime number.
The simplest finite fields, with prime order, are most directly accessible usingmodular arithmetic. For a fixed positive integern, arithmetic "modulon" means to work with the numbers
Z/nZ = {0, 1, ...,n − 1}.
The addition and multiplication on this set are done by performing the operation in question in the setZ of integers, dividing byn and taking the remainder as result. This construction yields a field precisely ifn is aprime number. For example, taking the primen = 2 results in the above-mentioned fieldF2. Forn = 4 and more generally, for anycomposite number (i.e., any numbern which can be expressed as a productn =r ⋅s of two strictly smaller natural numbers),Z/nZ is not a field: the product of two non-zero elements is zero sincer ⋅s = 0 inZ/nZ, which, as was explainedabove, preventsZ/nZ from being a field. The fieldZ/pZ withp elements (p being prime) constructed in this way is usually denoted byFp.
Every finite fieldF hasq =pn elements, wherep is prime andn ≥ 1. This statement holds sinceF may be viewed as avector space over its prime field. Thedimension of this vector space is necessarily finite, sayn, which implies the asserted statement.[15]
Such a splitting field is an extension ofFp in which the polynomialf hasq zeros. This meansf has as many zeros as possible since thedegree off isq. Forq = 22 = 4, it can be checked case by case using the above multiplication table that all four elements ofF4 satisfy the equationx4 =x, so they are zeros off. By contrast, inF2,f has only two zeros (namely0 and1), sof does not split into linear factors in this smaller field. Elaborating further on basic field-theoretic notions, it can be shown that two finite fields with the same order are isomorphic.[16] It is thus customary to speak ofthe finite field withq elements, denoted byFq orGF(q).
Historically, three algebraic disciplines led to the concept of a field: the question of solving polynomial equations,algebraic number theory, andalgebraic geometry.[17] A first step towards the notion of a field was made in 1770 byJoseph-Louis Lagrange, who observed that permuting the zerosx1,x2,x3 of acubic polynomial in the expression
(x1 +ωx2 +ω2x3)3
(withω being a thirdroot of unity) only yields two values. This way, Lagrange conceptually explained the classical solution method ofScipione del Ferro andFrançois Viète, which proceeds by reducing a cubic equation for an unknownx to a quadratic equation forx3.[18] Together with a similar observation forequations of degree 4, Lagrange thus linked what eventually became the concept of fields and the concept of groups.[19]Vandermonde, also in 1770, and to a fuller extent,Carl Friedrich Gauss, in hisDisquisitiones Arithmeticae (1801), studied the equation
xp = 1
for a primep and, again using modern language, the resulting cyclicGalois group. Gauss deduced that aregularp-gon can be constructed ifp = 22k + 1. Building on Lagrange's work,Paolo Ruffini claimed (1799) thatquintic equations (polynomial equations of degree5) cannot be solved algebraically; however, his arguments were flawed. These gaps were filled byNiels Henrik Abel in 1824.[20]Évariste Galois, in 1832, devised necessary and sufficient criteria for a polynomial equation to be algebraically solvable, thus establishing in effect what is known asGalois theory today. Both Abel and Galois worked with what is today called analgebraic number field, but conceived neither an explicit notion of a field, nor of a group.
In 1871Richard Dedekind introduced, for a set of real or complex numbers that is closed under the four arithmetic operations, theGerman wordKörper, which means "body" or "corpus" (to suggest an organically closed entity). The English term "field" was introduced byMoore (1893).[21]
By a field we will mean every infinite system of real or complex numbers so closed in itself and perfect that addition, subtraction, multiplication, and division of any two of these numbers again yields a number of the system.
In 1881Leopold Kronecker defined what he called adomain of rationality, which is a field ofrational fractions in modern terms. Kronecker's notion did not cover the field of all algebraic numbers (which is a field in Dedekind's sense), but on the other hand was more abstract than Dedekind's in that it made no specific assumption on the nature of the elements of a field. Kronecker interpreted a field such asQ(π) abstractly as the rational function fieldQ(X). Prior to this, examples of transcendental numbers were known sinceJoseph Liouville's work in 1844, untilCharles Hermite (1873) andFerdinand von Lindemann (1882) proved the transcendence ofe andπ, respectively.[23]
Acommutative ring is a set that is equipped with an addition and multiplication operation and satisfies all the axioms of a field, except for the existence of multiplicative inversesa−1.[26] For example, the integersZ form a commutative ring, but not a field: thereciprocal of an integern is not itself an integer, unlessn = ±1.
In the hierarchy of algebraic structures fields can be characterized as the commutative ringsR in which every nonzero element is aunit (which means every element is invertible). Similarly, fields are the commutative rings with precisely two distinctideals,(0) andR. Fields are also precisely the commutative rings in which(0) is the onlyprime ideal.
Given a commutative ringR, there are two ways to construct a field related toR, i.e., two ways of modifyingR such that all nonzero elements become invertible: forming the field of fractions, and forming residue fields. The field of fractions ofZ isQ, the rationals, while the residue fields ofZ are the finite fieldsFp.
Given anintegral domainR, itsfield of fractionsQ(R) is built with the fractions of two elements ofR exactly asQ is constructed from the integers. More precisely, the elements ofQ(R) are the fractionsa/b wherea andb are inR, andb ≠ 0. Two fractionsa/b andc/d are equal if and only ifad =bc. The operation on the fractions work exactly as for rational numbers. For example,
It is straightforward to show that, if the ring is an integral domain, the set of the fractions form a field.[27]
over a fieldF is the field of fractions of the ringF[[x]] offormal power series (in whichk ≥ 0). Since any Laurent series is a fraction of a power series divided by a power ofx (as opposed to an arbitrary power series), the representation of fractions is less important in this situation, though.
Theideal generated by a single polynomialf in the polynomial ringR =E[X] (over a fieldE) is maximal if and only iff isirreducible inE, i.e., iff cannot be expressed as the product of two polynomials inE[X] of smallerdegree. This yields a field
F =E[X] / (f(X)).
This fieldF contains an elementx (namely theresidue class ofX) which satisfies the equation
f(x) = 0.
For example,C is obtained fromR byadjoining theimaginary unit symboli, which satisfiesf(i) = 0, wheref(X) =X2 + 1. Moreover,f is irreducible overR, which implies that the map that sends a polynomialf(X) ∊R[X] tof(i) yields an isomorphism
Fields can be constructed inside a given bigger container field. Suppose given a fieldE, and a fieldF containingE as a subfield. For any elementx ofF, there is a smallest subfield ofF containingE andx, called the subfield ofF generated byx and denotedE(x).[29] The passage fromE toE(x) is referred to byadjoining an element toE. More generally, for a subsetS ⊂F, there is a minimal subfield ofF containingE andS, denoted byE(S).
Thecompositum of two subfieldsE andE′ of some fieldF is the smallest subfield ofF containing bothE andE′. The compositum can be used to construct the biggest subfield ofF satisfying a certain property, for example the biggest subfield ofF, which is, in the language introduced below, algebraic overE.[c]
The notion of a subfieldE ⊂F can also be regarded from the opposite point of view, by referring toF being afield extension (or just extension) ofE, denoted by
F /E,
and read "F overE".
A basic datum of a field extension is itsdegree[F :E], i.e., the dimension ofF as anE-vector space. It satisfies the formula[30]
[G :E] = [G :F] [F :E].
Extensions whose degree is finite are referred to as finite extensions. The extensionsC /R andF4 /F2 are of degree2, whereasR /Q is an infinite extension.
withen, ...,e0 inE, anden ≠ 0.For example, theimaginary uniti inC is algebraic overR, and even overQ, since it satisfies the equation
i2 + 1 = 0.
A field extension in which every element ofF is algebraic overE is called analgebraic extension. Any finite extension is necessarily algebraic, as can be deduced from the above multiplicativity formula.[31]
The subfieldE(x) generated by an elementx, as above, is an algebraic extension ofE if and only ifx is an algebraic element. That is to say, ifx is algebraic, all other elements ofE(x) are necessarily algebraic as well. Moreover, the degree of the extensionE(x) /E, i.e., the dimension ofE(x) as anE-vector space, equals the minimal degreen such that there is a polynomial equation involvingx, as above. If this degree isn, then the elements ofE(x) have the form
For example, the fieldQ(i) ofGaussian rationals is the subfield ofC consisting of all numbers of the forma +bi where botha andb are rational numbers: summands of the formi2 (and similarly for higher exponents) do not have to be considered here, sincea +bi +ci2 can be simplified toa −c +bi.
The above-mentioned field ofrational fractionsE(X), whereX is anindeterminate, is not an algebraic extension ofE since there is no polynomial equation with coefficients inE whose zero isX. Elements, such asX, which are not algebraic are calledtranscendental. Informally speaking, the indeterminateX and its powers do not interact with elements ofE. A similar construction can be carried out with a set of indeterminates, instead of just one.
Once again, the field extensionE(x) /E discussed above is a key example: ifx is not algebraic (i.e.,x is not aroot of a polynomial with coefficients inE), thenE(x) is isomorphic toE(X). This isomorphism is obtained by substitutingx toX in rational fractions.
A subsetS of a fieldF is atranscendence basis if it isalgebraically independent (do not satisfy any polynomial relations) overE and ifF is an algebraic extension ofE(S). Any field extensionF /E has a transcendence basis.[32] Thus, field extensions can be split into ones of the formE(S) /E (purely transcendental extensions) and algebraic extensions.
has a solutionx ∊F.[33] By thefundamental theorem of algebra,C is algebraically closed, i.e.,any polynomial equation with complex coefficients has a complex solution. The rational and the real numbers arenot algebraically closed since the equation
x2 + 1 = 0
does not have any rational or real solution. A field containingF is called analgebraic closure ofF if it isalgebraic overF (roughly speaking, not too big compared toF) and is algebraically closed (big enough to contain solutions of all polynomial equations).
By the above,C is an algebraic closure ofR. The situation that the algebraic closure is a finite extension of the fieldF is quite special: by theArtin–Schreier theorem, the degree of this extension is necessarily2, andF iselementarily equivalent toR. Such fields are also known asreal closed fields.
Any fieldF has an algebraic closure, which is moreover unique up to (non-unique) isomorphism. It is commonly referred to asthe algebraic closure and denotedF. For example, the algebraic closureQ ofQ is called the field ofalgebraic numbers. The fieldF is usually rather implicit since its construction requires theultrafilter lemma, a set-theoretic axiom that is weaker than theaxiom of choice.[34] In this regard, the algebraic closure ofFq, is exceptionally simple. It is the union of the finite fields containingFq (the ones of orderqn). For any algebraically closed fieldF of characteristic0, the algebraic closure of the fieldF((t)) ofLaurent series is the field ofPuiseux series, obtained by adjoining roots oft.[35]
Since fields are ubiquitous in mathematics and beyond, several refinements of the concept have been adapted to the needs of particular mathematical areas.
A fieldF is called anordered field if any two elements can be compared, so thatx +y ≥ 0 andxy ≥ 0 wheneverx ≥ 0 andy ≥ 0. For example, the real numbers form an ordered field, with the usual ordering ≥. TheArtin–Schreier theorem states that a field can be ordered if and only if it is aformally real field, which means that any quadratic equation
AnArchimedean field is an ordered field such that for each element there exists a finite expression
1 + 1 + ⋯ + 1
whose value is greater than that element, that is, there are no infinite elements. Equivalently, the field contains noinfinitesimals (elements smaller than all rational numbers); or, yet equivalent, the field is isomorphic to a subfield ofR.
Each bounded real set has a least upper bound.
An ordered field isDedekind-complete if allupper bounds,lower bounds (seeDedekind cut) and limits, which should exist, do exist. More formally, eachbounded subset ofF is required to have a least upper bound. Any complete field is necessarily Archimedean,[38] since in any non-Archimedean field there is neither a greatest infinitesimal nor a least positive rational, whence the sequence1/2, 1/3, 1/4, ..., every element of which is greater than every infinitesimal, has no limit.
Since every proper subfield of the reals also contains such gaps,R is the unique complete ordered field, up to isomorphism.[39] Several foundational results incalculus follow directly from this characterization of the reals.
ThehyperrealsR* form an ordered field that is not Archimedean. It is an extension of the reals obtained by including infinite and infinitesimal numbers. These are larger, respectively smaller than any real number. The hyperreals form the foundational basis ofnon-standard analysis.
Another refinement of the notion of a field is atopological field, in which the setF is atopological space, such that all operations of the field (addition, multiplication, the mapsa ↦ −a anda ↦a−1) arecontinuous maps with respect to the topology of the space.[40]The topology of all the fields discussed below is induced from ametric, i.e., afunction
d :F ×F →R,
that measures adistance between any two elements ofF.
Thecompletion ofF is another field in which, informally speaking, the "gaps" in the original fieldF are filled, if there are any. For example, anyirrational numberx, such asx =√2, is a "gap" in the rationalsQ in the sense that it is a real number that can be approximated arbitrarily closely by rational numbersp/q, in the sense that distance ofx andp/q given by theabsolute value|x −p/q| is as small as desired.The following table lists some examples of this construction. The fourth column shows an example of a zerosequence, i.e., a sequence whose limit (forn → ∞) is zero.
The fieldQp is used in number theory andp-adic analysis. The algebraic closureQp carries a unique norm extending the one onQp, but is not complete. The completion of this algebraic closure, however, is algebraically closed. Because of its rough analogy to the complex numbers, it is sometimes called the field ofcomplexp-adic numbers and is denoted byCp.[41]
finite extensions ofQp (local fields of characteristic zero)
finite extensions ofFp((t)), the field of Laurent series overFp (local fields of characteristicp).
These two types of local fields share some fundamental similarities. In this relation, the elementsp ∈Qp andt ∈Fp((t)) (referred to asuniformizer) correspond to each other. The first manifestation of this is at an elementary level: the elements of both fields can be expressed as power series in the uniformizer, with coefficients inFp. (However, since the addition inQp is done usingcarrying, which is not the case inFp((t)), these fields are not isomorphic.) The following facts show that this superficial similarity goes much deeper:
Anyfirst-order statement that is true for almost allQp is also true for almost allFp((t)). An application of this is theAx–Kochen theorem describing zeros of homogeneous polynomials inQp.
Adjoining arbitraryp-power roots ofp (inQp), respectively oft (inFp((t))), yields (infinite) extensions of these fields known asperfectoid fields. Strikingly, the Galois groups of these two fields are isomorphic, which is the first glimpse of a remarkable parallel between these two fields:[43]
Differential fields are fields equipped with aderivation, i.e., allow to take derivatives of elements in the field.[44] For example, the fieldR(X), together with the standard derivative of polynomials forms a differential field. These fields are central todifferential Galois theory, a variant of Galois theory dealing withlinear differential equations.
wheref is an irreducible polynomial (as above).[45] For such an extension, being normal and separable means that all zeros off are contained inF and thatf has only simple zeros. The latter condition is always satisfied ifE has characteristic0.
For a finite Galois extension, theGalois groupGal(F/E) is the group offield automorphisms ofF that are trivial onE (i.e., thebijectionsσ :F →F that preserve addition and multiplication and that send elements ofE to themselves). The importance of this group stems from thefundamental theorem of Galois theory, which constructs an explicitone-to-one correspondence between the set ofsubgroups ofGal(F/E) and the set of intermediate extensions of the extensionF/E.[46] By means of this correspondence, group-theoretic properties translate into facts about fields. For example, if the Galois group of a Galois extension as above is notsolvable (cannot be built fromabelian groups), then the zeros offcannot be expressed in terms of addition, multiplication, and radicals, i.e., expressions involving. For example, thesymmetric groupsSn is not solvable forn ≥ 5. Consequently, as can be shown, the zeros of the following polynomials are not expressible by sums, products, and radicals. For the latter polynomial, this fact is known as theAbel–Ruffini theorem:
f(X) =Xn +an−1Xn−1 + ⋯ +a0 (wheref is regarded as a polynomial inE(a0, ...,an−1), for some indeterminatesai,E is any field, andn ≥ 5).
Thetensor product of fields is not usually a field. For example, a finite extensionF /E of degreen is a Galois extension if and only if there is an isomorphism ofF-algebras
F ⊗EF ≅Fn.
This fact is the beginning ofGrothendieck's Galois theory, a far-reaching extension of Galois theory applicable to algebro-geometric objects.[48]
Basic invariants of a fieldF include the characteristic and thetranscendence degree ofF over its prime field. The latter is defined as the maximal number of elements inF that are algebraically independent over the prime field. Two algebraically closed fieldsE andF are isomorphic precisely if these two data agree.[49] This implies that any twouncountable algebraically closed fields of the samecardinality and the same characteristic are isomorphic. For example,Qp,Cp andC are isomorphic (butnot isomorphic as topological fields).
Inmodel theory, a branch ofmathematical logic, two fieldsE andF are calledelementarily equivalent if every mathematical statement that is true forE is also true forF and conversely. The mathematical statements in question are required to befirst-order sentences (involving0,1, the addition and multiplication). A typical example, forn > 0,n an integer, is
φ(E) = "any polynomial of degreen inE has a zero inE"
The set of such formulas for alln expresses thatE is algebraically closed.TheLefschetz principle states thatC is elementarily equivalent to any algebraically closed fieldF of characteristic zero. Moreover, any fixed statementφ holds inC if and only if it holds in any algebraically closed field of sufficiently high characteristic.[50]
IfU is anultrafilter on a setI, andFi is a field for everyi inI, theultraproduct of theFi with respect toU is a field.[51] It is denoted by
ulimi→∞Fi,
since it behaves in several ways as a limit of the fieldsFi:Łoś's theorem states that any first order statement that holds for all but finitely manyFi, also holds for the ultraproduct. Applied to the above sentenceφ, this shows that there is an isomorphism[e]
The Ax–Kochen theorem mentioned above also follows from this and an isomorphism of the ultraproducts (in both cases over all primesp)
ulimpQp ≅ ulimpFp((t)).
In addition, model theory also studies the logical properties of various other types of fields, such asreal closed fields orexponential fields (which are equipped with an exponential functionexp :F →F×).[52]
For fields that are not algebraically closed (or not separably closed), theabsolute Galois groupGal(F) is fundamentally important: extending the case of finite Galois extensions outlined above, this group governsall finite separable extensions ofF. By elementary means, the groupGal(Fq) can be shown to be thePrüfer group, theprofinite completion ofZ. This statement subsumes the fact that the only algebraic extensions ofGal(Fq) are the fieldsGal(Fqn) forn > 0, and that the Galois groups of these finite extensions are given by
Gal(Fqn /Fq) =Z/nZ.
A description in terms of generators and relations is also known for the Galois groups ofp-adic number fields (finite extensions ofQp).[53]
Algebraic K-theory is related to the group ofinvertible matrices with coefficients the given field. For example, the process of taking thedeterminant of an invertible matrix leads to an isomorphismK1(F) =F×.Matsumoto's theorem shows thatK2(F) agrees withK2M(F). In higher degrees, K-theory diverges from Milnor K-theory and remains hard to compute in general.
has a unique solutionx in a fieldF, namely This immediate consequence of the definition of a field is fundamental inlinear algebra. For example, it is an essential ingredient ofGaussian elimination and of the proof that anyvector space has abasis.[55]
The theory ofmodules (the analogue of vector spaces overrings instead of fields) is much more complicated, because the above equation may have several or no solutions. In particularsystems of linear equations over a ring are much more difficult to solve than in the case of fields, even in the specially simple case of the ringZ of the integers.
The sum of three pointsP,Q, andR on an elliptic curveE (red) is zero if there is a line (blue) passing through these points.
A widely applied cryptographic routine uses the fact that discrete exponentiation, i.e., computing
an =a ⋅a ⋅ ⋯ ⋅a (n factors, for an integern ≥ 1)
in a (large) finite fieldFq can be performed much more efficiently than thediscrete logarithm, which is the inverse operation, i.e., determining the solutionn to an equation
an =b.
Inelliptic curve cryptography, the multiplication in a finite field is replaced by the operation of adding points on anelliptic curve, i.e., the solutions of an equation of the form
A compact Riemann surface ofgenus two (two handles). The genus can be read off the field of meromorphic functions on the surface.
Functions on a suitabletopological spaceX into a fieldF can be added and multiplied pointwise, e.g., the product of two functions is defined by the product of their values within the domain:
For having afield of functions, one must consider algebras of functions that areintegral domains. In this case the ratios of two functions, i.e., expressions of the form
Thefunction field of an algebraic varietyX (a geometric object defined as the common zeros of polynomial equations) consists of ratios ofregular functions, i.e., ratios of polynomial functions on the variety. The function field of then-dimensionalspace over a fieldF isF(x1, ...,xn), i.e., the field consisting of ratios of polynomials inn indeterminates. The function field ofX is the same as the one of anyopen dense subvariety. In other words, the function field is insensitive to replacingX by a (slightly) smaller subvariety.
The function field is invariant underisomorphism andbirational equivalence of varieties. It is therefore an important tool for the study ofabstract algebraic varieties and for the classification of algebraic varieties. For example, thedimension, which equals the transcendence degree ofF(X), is invariant under birational equivalence.[56] Forcurves (i.e., the dimension is one), the function fieldF(X) is very close toX: ifX issmooth andproper (the analogue of beingcompact),X can be reconstructed, up to isomorphism, from its field of functions.[f] In higher dimension the function field remembers less, but still decisive information aboutX. The study of function fields and their geometric meaning in higher dimensions is referred to asbirational geometry. Theminimal model program attempts to identify the simplest (in a certain precise sense) algebraic varieties with a prescribed function field.
Global fields are in the limelight inalgebraic number theory andarithmetic geometry.They are, by definition,number fields (finite extensions ofQ) or function fields overFq (finite extensions ofFq(t)). As for local fields, these two types of fields share several similar features, even though they are of characteristic0 and positive characteristic, respectively. Thisfunction field analogy can help to shape mathematical expectations, often first by understanding questions about function fields, and later treating the number field case. The latter is often more difficult. For example, theRiemann hypothesis concerning the zeros of theRiemann zeta function (open as of 2017) can be regarded as being parallel to theWeil conjectures (proven in 1974 byPierre Deligne).
Cyclotomic fields are among the most intensely studied number fields. They are of the formQ(ζn), whereζn is a primitiventhroot of unity, i.e., a complex numberζ that satisfiesζn = 1 andζm ≠ 1 for all0 <m <n.[57] Forn being aregular prime,Kummer used cyclotomic fields to proveFermat's Last Theorem, which asserts the non-existence of rational nonzero solutions to the equation
xn +yn =zn.
Local fields are completions of global fields.Ostrowski's theorem asserts that the only completions ofQ, a global field, are the local fieldsQp andR. Studying arithmetic questions in global fields may sometimes be done by looking at the corresponding questions locally. This technique is called thelocal–global principle. For example, theHasse–Minkowski theorem reduces the problem of finding rational solutions of quadratic equations to solving these equations inR andQp, whose solutions can easily be described.[58]
Unlike for local fields, the Galois groups of global fields are not known.Inverse Galois theory studies the (unsolved) problem whether any finite group is the Galois groupGal(F/Q) for some number fieldF.[59]Class field theory describes theabelian extensions, i.e., ones with abelian Galois group, or equivalently the abelianized Galois groups of global fields. A classical statement, theKronecker–Weber theorem, describes the maximal abelianQab extension ofQ: it is the field
In addition to the additional structure that fields may enjoy, fields admit various other related notions. Since in any field0 ≠ 1, any field has at least two elements. Nonetheless, there is a concept offield with one element, which is suggested to be a limit of the finite fieldsFp, asp tends to1.[60] In addition to division rings, there are various other weaker algebraic structures related to fields such asquasifields,near-fields andsemifields.
There are alsoproper classes with field structure, which are sometimes calledFields, with a capital 'F'. Thesurreal numbers form a Field containing the reals, and would be a field except for the fact that they are a proper class, not a set. Thenimbers, a concept fromgame theory, form such a Field as well.[61]
Dropping one or several axioms in the definition of a field leads to other algebraic structures. As was mentioned above, commutative rings satisfy all field axioms except for the existence of multiplicative inverses. Dropping instead commutativity of multiplication leads to the concept of adivision ring orskew field;[g] sometimes associativity is weakened as well. The only division rings that are finite-dimensionalR-vector spaces areR itself,C (which is a field), and thequaternionsH (in which multiplication is non-commutative). This result is known as theFrobenius theorem. TheoctonionsO, for which multiplication is neither commutative nor associative, is a normedalternative division algebra, but is not a division ring. This fact was proved using methods ofalgebraic topology in 1958 byMichel Kervaire,Raoul Bott, andJohn Milnor.[62]
^The a priori twofold use of the symbol "−" for denoting one part of a constant and for the additive inverses is justified by this latter condition.
^Equivalently, a field is analgebraic structure⟨F, +, ⋅, −,−1, 0, 1⟩ of type⟨2, 2, 1, 1, 0, 0⟩, such that0−1 is not defined,⟨F, +, −, 0⟩ and are abelian groups, and⋅ is distributive over+.[10]
^Some authors also consider the fieldsR andC to be local fields. On the other hand, these two fields, also called Archimedean local fields, share little similarity with the local fields considered here, to a point thatCassels (1986, p. vi) calls them "completely anomalous".
^BothC andulimpFp are algebraically closed by Łoś's theorem. For the same reason, they both have characteristic zero. Finally, they are both uncountable, so that they are isomorphic.
^More precisely, there is anequivalence of categories between smooth proper algebraic curves over an algebraically closed fieldF and finite field extensions ofF(T).
^Historically, division rings were sometimes referred to as fields, while fields were calledcommutative fields.
Kiernan, B. Melvin (1971), "The development of Galois theory from Lagrange to Artin",Archive for History of Exact Sciences,8 (1–2):40–154,doi:10.1007/BF00327219,MR1554154,S2CID121442989
Kuhlmann, Salma (2000),Ordered exponential fields, Fields Institute Monographs, vol. 12, American Mathematical Society,ISBN0-8218-0943-1,MR1760173
Tits, Jacques (1957), "Sur les analogues algébriques des groupes semi-simples complexes",Colloque d'algèbre supérieure, tenu à Bruxelles du 19 au 22 décembre 1956, Centre Belge de Recherches Mathématiques Établissements Ceuterick, Louvain, Paris: Librairie Gauthier-Villars, pp. 261–289
