Use this tag for questions about fields and field theory in abstract algebra. A field is, roughly speaking, an algebraic structure in which addition, subtraction, multiplication, and division of elements are well-defined. Please use (galois-theory) instead for questions specifically about that ...

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3answers
37 views

Finding the fixed Field of $\sigma \in Aut(\mathbb{R}(t)/\mathbb{R})$

Finding the fixed Field of $\sigma \in Aut(\mathbb{R}(t)/\mathbb{R})$ Let $\sigma$ be such that $\sigma(t)=-t$. I assume there is only one automorphism like this, I am not sure exactly why... How ...
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1answer
26 views

Smallest field containing $F$ and $a \in K$

Definition. Given a field extension $K \supset F$ and an element $a \in K,$ define $F(a)$ to be the intersection of all subfields of $K$ that contain $F$ and $a.$ What is some more explicit notation ...
6
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3answers
129 views

Fields of arbitrary cardinality

So I took an introductory abstract algebra course a few semesters ago, and we were shown that groups and rings can both be made into products, i.e. if I have some group $G$ (resp. ring $R$) and some ...
3
votes
1answer
92 views

Show that $K = \mathbb{Q}(\sqrt{p} \ | \ \text{p is prime} \}$ is an algebraic and infinite extension of Q

Show that $K = \mathbb{Q}(\sqrt{p} \ | \ \text{p is prime} \}$ is a algebraic and infinite extension on Q. Well, if i consider for every p prime, the polynomial p(x)=x^2−p, then p(x) is in Q(p√∣p is ...
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1answer
18 views

What is a Galois closure and Galois group?

I was reading wikipedia for Galois groups and this term suddenly appears and there is no definition for it. What is a Galois closure of a field $F$? Does this mean a maximal Galois extension of $F$ ...
3
votes
1answer
19 views

The necessary and sufficient condition for a regular n-gon to be constructible by ruler and compass.

I have a problem concerning the necessary and sufficient condition for a regular n-gon to be constructible by ruler and compass. $\bf My$ $\bf question:$ For a given positive integer $n$, how can we ...
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0answers
17 views

Why is this a corollary of this theorem?

Lang - Algebra p.251 Proposition 6.11 Let $E/F$ be a normal field extension. Let $E^G$ be the fixed field of $Aut(E/F)$. Then, $E^G$ is purely inseparable over $F$ and $E$ is separable over ...
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1answer
15 views

Can a field extension still have “non-separability” above its maximal purely inseparable subextension?

Question 1 Let $E/F$ be an algebraic field extension. Let $K$ be the set of all elements of $E$ that are purely inseparable over $F$. Then, $E/K/F$ is a tower of fields, and $K/F$ is purely ...
2
votes
1answer
78 views

Roots of $f(x) = x^3+x^2-2x-1$

Roots of $f(x) = x^3+x^2-2x-1$ Show: $a_1=2\cos(\frac{2\pi}{7})$ is a root of $f$. [Edited here] $a_2 = a_1^2-2$ is a root of $f$. $a_3 = a_1^3-3a_1$ is a root of $f$. The first one is ...
2
votes
2answers
47 views

Is the degree of an infinite algebraic extensions always countable?

I guess this is right and try to prove it by using the fact that the polynomial ring $K[t]$ has a countable basis $1,x,x^2,\cdots$. But How to use this fact? Aside, if this statement is true. Is the ...
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0answers
44 views

Let $\mathbb{K} $ be a field of characteristic $p>0$ and $\mathbb{F} | \mathbb{K} $ a finite and separable extension.

Let $\mathbb{K}$ be a field of characteristic $p>0$ and $\mathbb{F}/ \mathbb{K}$ a finite and separable extension. Show that if $B=\{\alpha_1,\dots,\alpha_n\}$ is a basis, then ...
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1answer
26 views

Field Theory Problem in Beachy's Abstract Algebra involving field extensions and transcendental elements.

Let $\mathbb{F}=\mathbb{K}[u]$ where $u$ is transcendental over $\mathbb{K}$. Show that if $\mathbb{K} \subsetneq \mathbb{E} \subseteq \mathbb{F}$ then $u$ is algebraic over $E$. I'm guessing ...
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1answer
26 views

Transcendence bases [on hold]

Let $\Bbb k \subset \mathbb{K}$ be a field extension. Let $S_1,S_2 \subset \mathbb{K}$ be sets of algebraically dependent elements such every proper subset of $S_i$ is algebraically independent. I ...
4
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0answers
36 views

Existence of Jordan decomposition over finite field

Prove that over finite field $\mathbb F$ exists additive Jordan-Chevalley decomposition: for all matrix $M$ there are semisimple matrix $M_{s}$ and nilpotent matrix $M_{n}$ such that $M=M_{s}+M_{n}$. ...
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votes
1answer
26 views

Group under addition structure of a finite field of order 9 [on hold]

Suppose $(F,+,\cdot)$ is the finite field with 9 elements.Let $G=(F,+)$ and $H=(F\setminus\{0\},\cdot)$ denote the underlying additive and multiplicative groups, respectively. Then, (a) $G=(Z/3Z) ...
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0answers
19 views

What is a purely inseparable extension?

There are many different definitions of purely inseparable extension, and below is what I have chosen for my definition. (Since I don't know what is a standard one, if you know please tell me what ...
2
votes
4answers
93 views

Show that for $n \geq 2$, the $n^{th}$ cyclotomic polynomial is a reciprocal polynomial, i.e. $\Phi_{n}(x) = x^{\phi(n)}\Phi(n)(x^{-1})$.

Here $\phi(n)$ is the Euler totient function and the degree of $\Phi_{n}(x)$. What I've done so far: Let $\phi(n) = p$ so the following products each have p components. $$\Phi_{n}(x) = \prod_{k=1, ...
3
votes
3answers
43 views

for $n > 1$ : odd , prove that $\Phi_{2n}(x) = \Phi_{n}(-x)$

for $n > 1$ : odd , prove that $\Phi_{2n}(x) = \Phi_{n}(-x)$ Note: $\Phi_n(x)$ is the $n$th cyclotomic polynomial whose roots are the primitive $n$th roots of unity if n is odd then $-1$ cannot ...
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1answer
23 views

Roots of $X^{l-1}+1$ in a quadratic extension $F_q$, $q=l^2$

Consider a finite field $F_q$ where $q=l^2$ ($l$ can be of the form $p^m$). Does $F_q$ has a root of $X^{l-1}+1$? As $X^l+X = X(X^{l-1}+1)$ we can show that $X^{l-1}+1$ splits if it has a root. This ...
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votes
2answers
40 views

Finite fields, characteristics and the Fundamental Homomorphism Theorem

I am trying to make sense of this proposition. I am fine with part (a), for part (b) however, can you explain what the computation proves? Can you not verify a homomorphism by checking the 3 ...
3
votes
1answer
39 views

Algebraic closure for rings

Is there any notion of algebraic closure for commutative rings? I am specifically interested in such a concept for $\mathbb Z_n$, with $n$ not a prime (possibly square-free). Such a concept would be ...
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1answer
13 views

Showing polynomial is irreducible over field containing roots of unity.

Given a field $F$ containing all the roots of unity I'm trying to show that $f(x) = x^p - \alpha^p$ is irreducible over $F$ (where $\alpha$ is not in $F$). It's clear that $f$ splits in $F(\alpha)$ ...
4
votes
2answers
141 views

What is the splitting field of $x^3 - \pi$?

What is the splitting field of $x^3 - \pi$? Is it $\mathbb R(\sqrt[3] \pi, \xi_3)$ or $\mathbb Q(\sqrt[3] \pi, \xi_3)$? (where $\xi_3$ denotes the third root of unity) It is a polynomial over ...
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0answers
10 views

What is a separable closure? And why the standard definition of algebraic closure is defind as so?

Instead of defining it as a subfield of algebraic closure, what would be the natural way to define it? Here's what I figured out: Let $F$ be a field and $E$ be an extension field of $F$. ...
1
vote
1answer
26 views

Polynomial Rings

I know that an Euclidean ring is a principal ring and this last is a factorial ring. I also know that if $B$ is a field, then $B[x]$ is Euclidean. While, for instance $\mathbb{Z}[x]$ is not a ...
8
votes
1answer
213 views

Find intermediate fields of $\mathbb{Q}(\sqrt[3]{2}, \sqrt{3},i) \, | \, \mathbb{Q}(i)$

This is the problem I am facing: Compute the intermediate fields of the extension $K | \mathbb{Q}(i)$ where $K = \mathbb{Q}(\sqrt[3]{2}, \sqrt{3},i)$ and find the intermediate fields $M$ such ...
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0answers
24 views

Purely transcendental proper extension not algebraically closed? [on hold]

I'm having trouble proving this Dummit and Foote exercise: Prove that a purely transcendental proper extension of a field is never algebraically closed.
3
votes
1answer
17 views

Is normal extension algebraic?

Let $E/F$ be a field extension. Then $E/F$ is called normal iff there exists $\mathscr{A}\subset F[X]\setminus\{0\}$ such that $\forall f\in\mathscr{A}$, $f$ splits over $E$ and ...
3
votes
1answer
58 views

What is the degree of a real closure of an ordered field?

Given a subfield $F$ of $\Bbb R$, let the real-closure of $F$ be the smallest subfield of $\Bbb R$ which is real-closed and extends $F$. Since the theory of real-closed fields is complete, this is in ...
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votes
2answers
29 views

How to show $x,y,z \in A$ - Functions, Combinatorics

If $A \subseteq \{1,2,3,4,5,6\}$, how to show that for every $A$ there are $x,y,z \in \{1,2,3,4,5,6\}$, where $x,y,z$ can also be the same or at least not different from each other, and the following ...
4
votes
2answers
38 views

What will the underlying group of a field be isomorphic to?

Let $(F,+,.) $ be a finite field with 9 elements .,Let $G=(F,+)$ and $H=(F\setminus \{0\},.)$ denote the underlying additive and multiplicative groups .Thenwhat will $G$ and $H$ be isomorphic to:? WE ...
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vote
2answers
46 views

Field extension over $\mathbb{Q}$

I am given a subfield $E$ of $\mathbb{C}$ and asked to show that $[E : \mathbb{Q}] \le 10$ when every element of $E$ is a root of a polynomial in $\mathbb{Q}[x]$ of degree $10$. But I don't think ...
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0answers
21 views

Relations between galois group of polynomial and its factors

$f = f_1\dots f_n$ where $f_i$ is irreducible and distinct. What can i say about $\operatorname{Gal}(f/\mathbb{Q})$ if i know $\operatorname{Gal}(f_i/\mathbb{Q})$ for any(some) $i$.
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0answers
60 views

Galois group of $x^6-5x^3+6$

Let $f = x^6-5x^3+6$. I want to determine $\operatorname{Gal}(f/\mathbb{Q})$ without some group theory tricks (like Sylow's theorems) and without reduction $\bmod p$. Let $L_f = ...
2
votes
2answers
30 views

Galois group of $x^6-9$

$f = x^6-9 = (x^3-3)(x^3+3)$ Let $L_f$ be splitting field therefore $L_f = \mathbb{Q}[\sqrt[3]{3},e^{\frac{2\pi i}{3}}]$, $[L_f:\mathbb{Q}] = 9$. Also $Gal\space x^3±3/\mathbb{Q} = S_3$ and $Gal\space ...
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votes
0answers
17 views

Whether a given collection is a set or not [duplicate]

I originally knew that a set is a concept that has no definition. However, today, in abstract algebra class, the professor told us that the collection of all fields E such that E/F is an alegbraic ...
0
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1answer
20 views

Compositum of normal extensions is a normal extension

I'm trying to prove that if $ F \subset K, F \subset M $ are normal extensions, $ K,M \subset E $, then $ KM$ is also a normal extension of $ F $. I tried using the fact that $ F \subset KM $ is a ...
13
votes
9answers
358 views

Prove that $f=x^4-4x^2+16\in\mathbb{Q}$ is irreducible

Prove that $f=x^4-4x^2+16\in\mathbb{Q}[x]$ is irreducible. I am trying to prove it with Eisenstein's criterion but without success: for p=2, it divides -4 and the constant coefficient 16, don't ...
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3answers
42 views

Minimal polynomial of $\sqrt{2} + \sqrt{3}$ over $\Bbb{Q}(\sqrt{6})$

I have to find the minimal polynomial of $\alpha = \sqrt{2} + \sqrt{3}$ over $\Bbb{Q}(\sqrt6)$. $\alpha^{2} = 2 + 2\sqrt6 + 3$ so $f(X) = X^{2} - 5 - 2\sqrt6$ is a polynomial where $f(X) \in ...
0
votes
1answer
33 views

Degree of extension

How to find the degree of $\mathbb Q\left(\sqrt2+\sqrt[3]2\right) $ over $\mathbb Q\left(\sqrt2\right)$ ? I know how to find $\mathbb Q\left(\sqrt2\right)$ over $\mathbb Q$. But i am confused in ...
2
votes
1answer
22 views

Determine the Galois group of $ F(x^5) \subset F(x) $

I'm rather new to Galois theory and have been given this exercise: Suppose $ F $ is respectively equal to $ \mathbb{Q}, \mathbb{C}, \mathbb{F}_5 $ (the third one is just the 5-element field). My task ...
1
vote
1answer
17 views

Prove $\sigma_g(x) \in Aut(R(x)/R)$

Let $R$ be a field and let $R(x)$ be the field of rational functions in $x$ whose coefficients are in $R$. Let $g = \left( \begin{array}{cc} a & b \\ c & d \\ \end{array} \right) \in ...
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0answers
19 views

Subfields of splitting field of $x^3+x+1$

$f = x^3+x+1, L_f$ - splitting field of $f$. Discriminant $D = D(f) = -31$ therefore $\deg L_f/\mathbb{Q} = 6$ and $Gal L_f/\mathbb{Q} = S_3 $. I want to find all subfields of $L_f$. $L_f = ...
1
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0answers
14 views

Obtaining formula for roots of cubic equalation

$f = t^3+pt+q \in \mathbb{C}(p,q)$ I want prove that splitting field of $f$ is $$\mathbb{C}(p,q)[D,x]$$ $\mu_x = t^3-a$(over $\mathbb{C}(p,q)[D]$),$\mu_D = t^2-b$(over $\mathbb{C}(p,q)$). I think that ...
1
vote
1answer
22 views

Question on separable field extenions

Hi I was given this question which I cannot express myself mathematically on so would indeed like the help and appreciate it I am given $ K/F $ is a finite field extension. I am required to show that ...
2
votes
2answers
56 views

Elements in a field

Prove that there exists a field with $16$ elements. I know that there is a theorem that states that a finite field can only have $$ p^k $$ Where $p$ is a prime and $k$ is any positive integer. But ...
3
votes
3answers
53 views

In a commutative ring with identity, if $p$ is irreducible, is ($p$) a maximal ideal?

In a Euclidean Domain, $D$, if we mod out by an irreducible, $p$, we get the field $D/(p)$. I can see that this follows since we are going to be able to write $1$ as a linear combination of $p$ and ...
0
votes
1answer
37 views

Let E be an extension of a field F of degree 2. Show there is an element with $\beta = x^2 - \alpha$

Let E be an extension of a field F of degree 2. Show there is an element with $\beta = x^2 - \alpha$. I wanna know if my approach is correct. If it's an extension of degree 2, then $F(\alpha)$ ...
1
vote
1answer
43 views

Field extensions and algebraic elements

Can somebody explain why taking beta gives $K(\beta)$ as a subspace of $K(\alpha)$?
3
votes
1answer
54 views

Working in $\mathbb Q[x]$. Two polynomials are coprime if their gcd is a constant?

When are two polynomials coprime? Is it when their gcd is a constant? If we divide $x^3-7x-5$ by $x-4$, we get: $$x^3-7x-5=(x-4)(x^2+4x+9)+31$$ So, is $31$ their gcd, but since $31$ is not monic ...