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Question 1

0votes

0answers

38views

Prove that $\mathbb{Q}(\alpha, \beta, \gamma, r_1) = \mathbb{Q}(r_1, r_2, r_3, r_4)$

Problem Statement: Let $f(x) = x^4 + ax^3 + bx^2 + cx + d$ be an irreducible polynomial over the field of rational numbers $\mathbb{Q}$, where $a, b, c, d \in \mathbb{Q}$. Let $r_1, r_2, r_3, r_4$ be ...

HGF

1,017

asked1 hour ago

Question 2

1vote

0answers

103views

Splitting field of a polynomial over $\mathbb{Q}_p$?

How to find the degree and the ramification index of the splitting field of a certain polynomial over $\mathbb{Q}_p$? For instance, $f(x)=3+9x+3x^4+x^6$ over $\mathbb{Q}_3$?If $\gamma$ is one of the ...

8k14

311

askedNov 19 at 13:31

Question 3

1vote

1answer

95views

Show that $\{E:F\}\leq [E:F]$

I am doing the exercises of Fraleigh's book. Exercise 49, question 14.Let $F$ be a field, $E$ be a finite extension of $F$. Let $\{E:F\}$ be the number of isomorphisms of $E$ to a subfield of $\bar F$...

Brightsun

416

askedNov 17 at 12:44

Question 4

4votes

1answer

104views

Quadratic fields in $\mathbb{Q}(p^{\frac{1}{n}})$ where $n$ is even,p is prime.

I want to prove that there is a unique quadratic field extension in $\mathbb{Q}(p^{\frac{1}{n}})$, where $n = 2k$ and p is prime. Clearly $\mathbb{Q}(p^{\frac{1}{2}})$ $\subset$ $\mathbb{Q}(p^{\frac{1}...

Naveen Kumar

183

askedOct 21 at 11:54

Question 5

0votes

0answers

88views

Minimality condition for splitting field

Let $K$ be a field and let $p(t) = a_n t^n + \dots + a_0 \in K[t]$ be an irreducible polynomial. A splitting field of $p(t)$ is a field $L$ such that (1) $p(t) = a_n (t - \alpha_1) \dots (t - \alpha_n)...

hdecristo

1,265

askedOct 13 at 20:23

Question 6

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 7

0votes

1answer

86views

Splitting of reduced polynomial in residue field

This question comes from a proof in Neukirch's Algebraic Number Theory. Let $\mathcal{o}$ a Dedekind domain, $K$ its field of fractions, $L/K$ a Galois extension and $\mathcal{O}$ the integral closure ...

user1686844

45

askedSep 8 at 21:16

Question 8

3votes

1answer

143views

Why Gallian didn't consider the roots of unity for the splitting field of $x^6-2$?

I am following the chapter 20 (Extension Fields) of Gallian's book.I am confused with the computation of two splitting fields of two different polynomials $x^6-2$ and $x^n-a$, both over $\mathbb{Q}$. ...

Learner

574

askedSep 2 at 3:22

Question 9

-3votes

2answers

118views

Elaborated example of finite field construction (characteristic 2) [closed]

I read the Wikipedia on finite field and cannot understand several explicit construction steps. I want an elaborated explanation on construction of $GF(4)$, $GF(8)$, $GF(16)$ fields, that answers on ...

unennhexium

1

askedJul 22 at 8:34

Question 10

2votes

1answer

113views

Compute the cardinal number of the splitting field over a finite field

Let $F$ be the splitting field of $f = X^{10}-1 \in \mathbb{F}_{3}[X]$. Determine the cardinality of $F$.At first I tried to brute force by finding out an irreducible factor. I got to the ...

MrGran

1,004

askedJul 10 at 15:37

Question 11

1vote

1answer

83views

Splitting of the cyclotomic polynomials over finite fields

I'm trying to show thatLet $p$ be a prime and $a$ be such that $\gcd(a,p) = 1$. The cyclotomic polynomial $\Phi_a(x)$ splits into linear factors over $\mathbb F_p$ if and only if $p\equiv 1 \pmod a$...

Robi

117

askedJun 12 at 15:35

Question 12

1vote

2answers

143views

Degree of the minimal polynomial over a finite field.

Let $F= \mathbb{F}_{3}(a)$ a field with $a$ verifying the equation $a^{3}+a^{2}-1 = 0$.Determine the cardinality of $F$.Determine the degree of $Irr(a^{2}, \mathbb{F}_{3})$Compute $Irr(a^{2}, \...

MrGran

1,004

askedJun 5 at 18:26

Question 13

0votes

1answer

89views

How can I find the minimal polynomial over a finite field in a different way?

Let $F$ be the splitting field of $f = x^{2}+x+1 \in \mathbb{F}_{5}[x]$ and let $a \in F$ be a root of $f$. Prove that the polynomial $g = Irr(a+1, \mathbb{F}_{5})$ verifies $g(a) = 3a$.We first show ...

MrGran

1,004

askedJun 4 at 17:09

Question 14

0votes

0answers

67views

Clarifying Cohn's proof of algebraic closure existence

I'm struggling to understand the following proof in Cohn's Algebra, Vol. 2, of the "existence" part of his Prop. 3.3: "For every set $F$ of polynomials over a field $k$ there is a ...

WillG

7,769

askedMay 30 at 20:01

Question 15

3votes

1answer

174views

Splitting field of $x^5 - 20$ over $\mathbb Q$ is not generated by sum of two distinct roots

This is a review problem for an algebra exam:Let $\alpha, \beta$ be distinct roots of the polynomial $f(x)=x^5 - 20 \in \mathbb Q[x]$, and let $K$ be the splitting field of $f(x)$.Show that $K= \...

algebra learner

591

askedApr 28 at 17:34

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Questions tagged [splitting-field]

Prove that $\mathbb{Q}(\alpha, \beta, \gamma, r_1) = \mathbb{Q}(r_1, r_2, r_3, r_4)$

Splitting field of a polynomial over $\mathbb{Q}_p$?

Show that $\{E:F\}\leq [E:F]$

Quadratic fields in $\mathbb{Q}(p^{\frac{1}{n}})$ where $n$ is even,p is prime.

Minimality condition for splitting field

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

Splitting of reduced polynomial in residue field

Why Gallian didn't consider the roots of unity for the splitting field of $x^6-2$?

Elaborated example of finite field construction (characteristic 2) [closed]

Compute the cardinal number of the splitting field over a finite field

Splitting of the cyclotomic polynomials over finite fields

Degree of the minimal polynomial over a finite field.

How can I find the minimal polynomial over a finite field in a different way?

Clarifying Cohn's proof of algebraic closure existence

Splitting field of $x^5 - 20$ over $\mathbb Q$ is not generated by sum of two distinct roots

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