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Question 1

0votes

0answers

50views

Function fields over finite field.

Suppose $\mathbb{F}_q$ is a finite field of characteristic $p>0$ and $t$ is transcendental over $\mathbb{F}_q$. Then is this field $\mathbb{F}_q(t)$ a Hilbertian field?The definition of ...

Math123

1,273

asked2 days ago

Question 2

2votes

1answer

223views

Cyclicity of the multiplicative group of the integers modulo a prime

Edit. It seems that the Lemma 2 needs already the existence of a primitive root modulo $p$. If there's no other way to prove it, then my argument is pointless.(NB: I'm aware that there's plenty of ...

Kan't

5,601

askedNov 26 at 9:47

Question 3

2votes

1answer

96views

Automorphism group of $UT(3,p)$

Here $p>2$ is a prime, the group $UT(3,p)$ is the group of $3\times3$ upper unitriangular matrices with coefficients in $\mathbb{F}_p$:$$\begin{bmatrix}1&x&y\\0&1&z\\0&0&...

tanjia

151

askedNov 25 at 12:28

Question 4

2votes

1answer

80views

Counting disjoint $k$-tuples of lines in $\mathbb F_q^n$

Let $k$ be a positive integer. How many $k$-tuples of disjoint lines are there in $\mathbb F_q^n$? Here two lines are disjoint if they do not share a point in $\mathbb F_q^n$.I was wondering because ...

popop614

111

askedNov 22 at 5:23

Question 5

2votes

0answers

49views

Conceptual Explanation for Gauss Periods Normal Basis Criterion

I am studying the construction of normal bases using Gauss periods. Let $r = nk + 1$ be prime, $q$ a prime power with $\gcd(q, r) = 1$, and $\mathcal{K}$ the unique subgroup of $\mathbb{Z}_r^\times$ ...

Shirogane

175

askedNov 12 at 19:57

Question 6

5votes

1answer

228views

Sufficient conditions for representing the matrice $U$ as $B^\top B^{-1}$

Out of curiosity, I came across the study of matrices of the form $U = B^\top B^{-1}$ for $B \in GL(n,\mathbb{F}_2)$. In essence, U represents how matrice $B$ is asymetric.This raised the question: ...

pvrnothere

61

askedNov 5 at 10:07

Question 7

2votes

1answer

217views

Is there an interpretation of why vectors can be orthogonal to themselves in finite fields?

I'm learning linear algebra through Philip Klein's Coding the Matrix, and he defines the annihilator of a vector subspace $V\subset \mathbb{F}^n$ as $V^0=\{\mathbf{u}\in\mathbb{F}^n:\mathbf{u}\cdot\...

Vitor

21

askedOct 30 at 0:49

Question 8

3votes

1answer

179views

What is the significance of studying 2-torsion points in Elkies' work on supersingular elliptic curves?

I'm studying Elkies paper about the infinity of supersingular primes for every elliptic curve over $\mathbb{Q}$. I'm having some trouble finding out why Elkies studies in particular the 2-torsion ...

Samuel F.

31

askedOct 12 at 22:13

Question 9

1vote

1answer

143views

Generators in the multiplicative group of the field $\mathbb{F}_{16}$ [duplicate]

The following are two theorems about finite field:Theorem 1. Any finite field $\mathbb{F}_{p^n}$ is isomorphic to the quotient ring $\mathbb{F}_p[x] / (g(x))$, where $g(x)$ is an irreducible ...

AkDumb

41

askedOct 10 at 14:16

Question 10

3votes

1answer

372views

Degree of splitting field of $X^n-a$ over $\mathbb{F}_p$

Factoring polynomials over finite fields can be done by the Cantor–Zassenhaus algorithm (see this Wikipedia article for a brief description).So by factorization algorithms over $\mathbb{F}_p$, we can ...

Afntu

4,285

askedOct 6 at 18:45

Question 11

2votes

1answer

167views

Let $q=2^k$ with $k$ even. Let $F=GF(q^3)$ and $G=\{x \in F ~|~ x^{q^2+q+1}=1\}$. If $x+x^q+x^{q^2}=1$ for $x \in G$,then $x=t^3$ for some $t\in G$.

Let $q=2^k$ with $k$ even. Let $F=GF(q^3)$ and $G=\{x \in F ~| ~x^{q^2+q+1}=1\}$. For $x\in G$, if $x+x^q+x^{q^2}=1$, then $x=t^3$ for some $t\in G$. Numerical data for k=2,4,6,8 indicate the above ...

Sunshning

163

askedOct 1 at 2:57

Question 12

0votes

0answers

71views

Circular convolution modulo $3$

I am working with a convolution sum of the form$$ h(j) = \sum_{k=0}^2 f\!\big((j-k) \bmod 3\big)\, g(k), $$where $f, g : \{0,1,2\} \to \mathbb{C}$. Because of the modulo $3$ structure in the index ...

AmB

27

askedSep 30 at 1:28

Question 13

2votes

1answer

97views

Basis for algebraic closure of $\mathbb{F}_2$

For any finite degree extension $\mathbb{F}_{2^t}$ of $\mathbb{F}_2$, we have at least a couple of canonical bases -- one being $\{1,\gamma,\ldots,\gamma^{t-1}\}$ where $\gamma$ is a primitive element,...

The Discrete Guy

45

askedSep 29 at 14:16

Question 14

2votes

0answers

44views

Linear recurrence for the number of elements of a finite field with given trace and subtrace

Trace and subtrace of an element $a$ of a finite field $\mathrm{GF}(p^n)$ are the cofficients of respectively $x^{n-1}$ (negated) and $x^{n-2}$ in the characteristic polynomial$$ p_a(x) = (x-a)(x-a^p)...

colt_browning

3,770

askedSep 28 at 9:37

Question 15

-4votes

1answer

134views

What is the kernel of the Frobenius mapping from $\mathbb{F}_{p^n}$ to itself?? [duplicate]

The Frobenius map from $\mathbb{F}_{p^n}$ to itself is a ring automorphism of order $n$.I am confused how to show it is injective.For, if $\phi :\mathbb{F}_{p^n} \to \mathbb{F}_{p^n}$ is defined by $...

Learner

574

askedSep 22 at 6:15

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Stack Exchange Network

Questions tagged [finite-fields]

Function fields over finite field.

Cyclicity of the multiplicative group of the integers modulo a prime

Automorphism group of $UT(3,p)$

Counting disjoint $k$-tuples of lines in $\mathbb F_q^n$

Conceptual Explanation for Gauss Periods Normal Basis Criterion

Sufficient conditions for representing the matrice $U$ as $B^\top B^{-1}$

Is there an interpretation of why vectors can be orthogonal to themselves in finite fields?

What is the significance of studying 2-torsion points in Elkies' work on supersingular elliptic curves?

Generators in the multiplicative group of the field $\mathbb{F}_{16}$ [duplicate]

Degree of splitting field of $X^n-a$ over $\mathbb{F}_p$

Let $q=2^k$ with $k$ even. Let $F=GF(q^3)$ and $G=\{x \in F ~|~ x^{q^2+q+1}=1\}$. If $x+x^q+x^{q^2}=1$ for $x \in G$,then $x=t^3$ for some $t\in G$.

Circular convolution modulo $3$

Basis for algebraic closure of $\mathbb{F}_2$

Linear recurrence for the number of elements of a finite field with given trace and subtrace

What is the kernel of the Frobenius mapping from $\mathbb{F}_{p^n}$ to itself?? [duplicate]

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