Galois theory allows one to reduce certain problems in field theory, especially those related to field extensions, to problems in group theory. For questions about field theory and not Galois theory, use the (field-theory) tag instead. For questions about abstractions of Galois theory, use ...

learn more… | top users | synonyms

1
vote
2answers
50 views

Galois Group of the extension $\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5}):\mathbb{Q}$. Why is it like this?

The $4$th edition of Ian Stewart's Galois Theory is what I am looking at where there is an example stated in the following manner Let the field extension be ...
1
vote
1answer
37 views

Minimal polynomial of generating Galois automorphism

Let $K/k$ be a finite cyclic Galois extension with Galois group generated by $\sigma$. Now forget about the $k$-algebra structure and interpret $\sigma$ as an endomorphism of the underlying $k$-vector ...
2
votes
0answers
37 views

Having trouble with Hungerford V.7 proof [duplicate]

Let $K$ be a field, $\bar{K}$ an algebraic closure and $\sigma \in \mathrm{Aut}_K \bar{K}$. Let $F=\{u \in \bar{K} \mid \sigma(u)=u\}$. Then prove $F$ is a field and every finite-dimensional extension ...
0
votes
2answers
56 views

Constructing Finite Field Tables

Could someone please walk me through how to construct Finite Field tables? My biggest confusion is how to get the elements from the respectable fields. For example. I'm asked to construct the table ...
0
votes
0answers
23 views

How to find all the subfields of $\mathbb{F}_{p^r}$ [duplicate]

How does one find all the subfields of the finite field $$\mathbb{F}_{p^r}$$ where $p$ is a prime?
5
votes
2answers
53 views

Is there a way to find one irreducible polynomial of degree n in the field Z2

I'm asked to find an irreducible polynomial of a certain degree in the field Z2. I won't specify the degree I'm asked here because I'd like to understand the method and know how to apply it to later ...
2
votes
1answer
23 views

Showing that the galois group for x^n+ax+b is doubly transitive

I read here that if $\alpha$ is a root of $x^n+ax+b $ and $\frac{x^n-\alpha^n}{x-\alpha}=a$ is irreducible, then the Galois group $G$ is doubly transitive. I don't think I understand the reasoning ...
3
votes
2answers
104 views

Computing normal closure and Galois group of quintic $x^5 - 3$

Having trouble big time. I am asked to find the normal closure for the extension $\mathbb{Q}(a):\mathbb{Q}$ where $a$ is the fifth root of $3$ and real. Then I am asked to find Galois groups for ...
1
vote
1answer
45 views

Does every automorphism of a field containing the rationals fix the rationals?

Every automorphism $f$ fixes $1$, so every automorphism fixes the integers, and we must have $f(p/q) = f(p)/f(q) = p/q$ for $p$, $q$ integers with $q$ nonzero. This is right, right? Also, does it ...
2
votes
4answers
71 views

What is a Galois Group here and why?

I am confused as to why the Galois group is as follows for my problem. Find the splitting field for $f(x)=x^2+1$ and find the Galois group $Gal(f)$. Now, The splitting field is $\Sigma= ...
3
votes
3answers
75 views

Question on a permutation of the roots of a Galois number field, dihedral over the rationals

The polynomial $f(x) = x^6 - 7x^5 + 21x^4 - 41x^3 + 63x^2 - 63x + 27$ defines a Galois extension $H$ of $\mathbb{Q}$. The Galois group of the extension $H/\mathbb{Q}$ is dihedral, and depending on ...
1
vote
0answers
23 views

Discriminant of minimal polynomial of an elementary symmetric function of roots

Through various examples I have noticed the following result which seems to be true. I do not see how to prove it, nor how to find counterexamples. Let $P(X)$ be a monic irreducible polynomial with ...
1
vote
1answer
43 views

Field extensions and degree; is my reasoning correct?

I am looking at a problem and wondering something... here is the question If $K \subseteq L$ are fields, show that for any $a$, that $[L(a):L] \leq [K(a):K]$ Now, one thing that bothers me; is ...
2
votes
3answers
42 views

When is $i$ contained in $Q(\zeta)$

As part of a problem I have to determine for which values of $n$ is $i$ contained in $\Bbb Q(\zeta)$, were $\zeta$ is a primitive $n^{th}$ root of the unity. Clearly, if $n$ is multiple of $4$, it is ...
1
vote
1answer
39 views

Trace of a function on an elliptic curve

Let $K/F$ be a Galois extension of number fields with Galois group $G$. Let $E$ be an elliptic curve defined over $F$ and $f \in K(E)^{\times}$ be a function. Define the trace of $f$ to be ...
1
vote
0answers
54 views

Field Extension Embedding into Matrix Conditions

Let $L/K$ be a field extension. How do we prove $L$ is a subfield of $M_n(K)$ (n by n matrices with entries in $K$) if and only if $[L:K]\mid n$? My attempt: I can't prove the forward direction. For ...
0
votes
1answer
26 views

Galois group of $x^p-a$ is isomorphic to a subgroup of Sym$(\mathbb{Z}/p\mathbb{Z}$)

This is a question from BAI, Jacobson: Assume that $x^p-a \in \mathbb{Q}[x]$ is irreducible. Then the Galois group of $x^p-a$ over $\mathbb{Q}$ is isomorphic to a subgroup of Sym($\mathbb{Z}_p$) of ...
3
votes
1answer
35 views

How can we show that all the roots of some irreducible polynomial are not of algebriacally equal status?

In studying Galois theory, I found that all roots of some irreducible polynomial are not of algebraically equal status, because the Galois group of some irreducible polynomial may not be full ...
0
votes
1answer
25 views

Given $\sigma,\tau\in\operatorname{Gal}(L/K)$, is there an $\alpha\in L$ such that $\sigma\alpha=\alpha$ and $\tau\alpha\neq\alpha$?

Let $L/K$ be a Galois extension with group $G$. Given two elements $\sigma,\tau\in G$, Is there an element $\alpha\in L$ such that $\sigma(\alpha)=\alpha$ and $\tau(\alpha)\neq\alpha$? This is not ...
0
votes
0answers
15 views

Infinite Galois theory: every subgroup of finite index is open. Proof-check

Let $\Omega/k$ be a (possibly infinite) Galois extension and let the group $G=G(\Omega/k)$ be equipped with the Krull Topology. My question is about a statement I think its true, but I'm not sure. ...
2
votes
1answer
30 views

The difference of each roots of some irreducible polynomial.

Let $E/F$ be a finite Galois extension and $\alpha\in E$ and $p(t)=irr(\alpha,F)$, the monic irreducible polynomial defined over $F$ which has a root $\alpha$. Let $\beta\in E$ be another root of ...
4
votes
0answers
32 views

How to tell if a set is open in the Krull topology?

I'm an undergraduate not very familiar with topology trying to understand the so called Krull Topology in the context of infinite Galois Theory. We proceed as follows: Let $\Omega/k$ be a (possibly ...
0
votes
1answer
41 views

To show that $\mathbf Q(\sqrt[r]{p_1}, \ldots, \sqrt[r]{p_n})=\mathbf Q(\sqrt[r]{p_1}+\cdots+\sqrt[r]{p_n})$

In this thread @Geoff Robinson gives a nice argument to show that if $p_1, \ldots, p_n$ are distinct primes then we have $\mathbf Q(\sqrt{p_1}, \ldots, \sqrt{p_n})=\mathbf ...
1
vote
2answers
52 views

Finding isomorphisms between finite fields.

I'm having trouble understanding how to find isomorphisms between finite fields. In my lecture notes it uses the following theorem: A function $f$ is an isomorphism from $GF(z^n)$ represented ...
0
votes
2answers
42 views

Primitive element of the extension $\mathbb Q(\sqrt{2},\sqrt{3})$ over $\mathbb Q$ [duplicate]

The title says it. I want to find an element $\alpha$ such that $\mathbb Q(\alpha)=\mathbb Q(\sqrt{2},\sqrt{3})$. I tried something like $\sqrt{2}+\sqrt{3}$ but that didn't help...
2
votes
1answer
55 views

Algebraic closure of a perfect field.

I don't know if this result is true or not, if we are in the first case, how can I prove it ? $$k \subset \overline{k} \text{ is Galois Extension } \Leftrightarrow k \text{ is a perfect field} $$
2
votes
1answer
45 views

The degree of a splitting field of a polynomial proof

I see this theorem in several places Let F be a field and $f(x)\in F[x]$ have degree $n\geq2$.Let $E/F$ be a splitting field of $f(x)$. Then $[E:F]$ divides $n!$ The proof involves some induction. ...
1
vote
0answers
36 views

Simple Field extensions.

Suppose $[L:K]=p$ $p$=prime .Then there exists $u$ such that $L=K(u)$ Proof.If $[L:K]=p$ $p$=prime then for $M$ an arbitrary field extension must hold that $$[L:K]=[L:M][M:K]$$ if $[L:K]=prime$ ...
3
votes
6answers
111 views

$x^4 -10x^2 +1 $ is irreducible over $\mathbb Q$

I have seen the thread Show that $x^4-10x^2+1$ is irreducible over $\mathbb{Q}$ but this didn't really have a full solution. Is it true that if it is reducible then it can be factored into a linear ...
1
vote
2answers
78 views

Linear independence of $\{1,\sqrt2,\sqrt3,\sqrt6 \}$ over $\mathbb Q$ [duplicate]

We want to prove that $a+b\sqrt2+c\sqrt3 +d\sqrt6 =0$ where $a,b,c,d \in \mathbb Q$ such that $a=b=c=d=0$. Can we prove that there exists no linear combination of $1, \sqrt3, \sqrt6$ that gives ...
2
votes
2answers
106 views

Basis for $\mathbb Q (\sqrt2 , \sqrt3 )$ over $\mathbb Q$

List a basis for $\mathbb K =\mathbb Q (\sqrt2 , \sqrt3 )$ as a vector space over $\mathbb Q $. I don't know how people come to the conclusion of a claimed basis. Like I am pretty sure that we just ...
2
votes
1answer
27 views

If $E$ is the Splitting Field for an Irreducible Separable Polynomial over $F$ then $[E:F]=\text{Aut}(E:F)$

$\DeclareMathOperator{\aut}{Aut}$ $\newcommand{\vp}{\varphi}$Let $p(x)$ be an irreducible and separable polynomial over a field $F$ and let $E$ be the splitting field for $p(x)$ over $F$. ...
0
votes
1answer
52 views

Show $\min _ {a, \mathbb F }= \frac1c f$

Suppose $\mathbb K / \mathbb F $ is an arbitrary finite field extension. Prove that if $f \in \mathbb F [x]$ is irreducible of degree at least $1$ and $a \in \mathbb K $ is a root of $f$ then $\min _ ...
2
votes
1answer
66 views

On calculating the restriction of the ideals in polynomial rings.

Let $K$ be a field and $F$ a Galois extension of $K$ and $G$ the Galois group of extension $F/K$. For any $\sigma\in G$, define an $K$-algebra automorphism $\sigma^*$ on $F[X_1,\dots,X_n]$ by ...
3
votes
3answers
121 views

Find the minimal polynomial of $\sqrt2 + \sqrt3 $ over $\mathbb Q$

I have no idea how to do this. To find the minimal polynomial of say $\sqrt2 + \sqrt3$, we need to find the monic polynomial $p \in \mathbb Q$ (correct if I am wrong but monic polynomial is when the ...
1
vote
3answers
101 views

Prove $\mathbb Q (\sqrt2 + \sqrt3 ) = \mathbb Q (\sqrt2 , \sqrt3 )$ [duplicate]

So correct if I am wrong, but these sets are: $\mathbb Q (\sqrt2 + \sqrt3 )=\{a+b\sqrt2 + c\sqrt3 +d \sqrt6 : a,b,c,d \in \mathbb Q \}$ $\mathbb Q (\sqrt2 , \sqrt3 )=\{a+b\sqrt2 + c\sqrt3 : a,b,c ...
3
votes
2answers
52 views

Minimal polynomial of cyclic $p^{n}$ extension of $\mathbb{Q}$?

Excuse me if the following question is stupid: When $p$ is an odd prime, we have $\operatorname{Gal}(\mathbb{Q}(\mu_{p^{n+1}})/\mathbb{Q})=\mathbb{Z}/(p-1)\mathbb{Z}\times\mathbb{Z}/p^{n}\mathbb{Z}$. ...
1
vote
1answer
26 views

Union of infinite subfields of the complex plane

Is the union of all subfields $\mathbb{Q}(\sqrt{n})$ for $n \geq 1$ a subfield of $\mathbb{C}$? The previous section of the question asked me to prove that the union of an infinite number of ...
3
votes
0answers
79 views

Prove $\sqrt3 \notin \mathbb Q (\sqrt2)$

Let $\mathbb F = \mathbb Q (\sqrt2)$ and $\mathbb K = \mathbb Q(\sqrt2, \sqrt3 )$. Prove $\sqrt3 \notin \mathbb Q (\sqrt2)$. I know that $\mathbb Q (\sqrt2)=\{a+b \sqrt2 : a,b \in \mathbb Q\}$. $ ...
0
votes
1answer
46 views

Solvability of polynomial equations of degree 4 (see question at the end)

Let $f$ be irreducible and separable over $k$ and $\text{char}(k) \neq 2, 3$. Further let $f(x) = x^4 +px^2 +qx + r = (x-\alpha_1)(x-\alpha_2)(x-\alpha_3)(x-\alpha_4)$ which splits over $K$. Then: ...
7
votes
4answers
727 views

Determining a matrix from its characteristic polynomial

Let $A\in\mathcal{M}_{n}(K)$, where $K$ is a field. Then, we can obtain the characteristic polynomial of $A$ by simply taking $p(\lambda)=\det(A-\lambda I_n)$, which give us something like ...
0
votes
3answers
86 views

Why is $\text{Gal}(K/k) \cong \text{Gal}(K(\omega)/k(\omega))$ with $\omega \notin K$?

The situation is as follows: $K$ is the splitting field of $\; x^3+px+q \in k[x]$ and $ K(\omega)$ is the field over which $\; x^3-1$ splits. That's probably one of those things you just see, but I ...
1
vote
1answer
54 views

What are the invariants of a number field? [closed]

How is defined an invariant of a field? Given a certain field extension $L/K$, is it related with the Galois group ${\rm Gal}(L/K)$? In the case of number fields, which are the invariants associated ...
1
vote
1answer
38 views

Expected genus of a function field over a finite field

Let $K=\mathbb F_q(t)$ be the field of rational functions over $\mathbb F_q$ and $f(T)\in K[T]$ be an irreducible polynomial. Let $F=K[y]/f(y)$ and $E$ be the Galois closure of F/K. What should I ...
1
vote
1answer
25 views

Abelianization of the absolute group and maximal abelian extension

Let $K$ be any field, $\overline K$ is the separable closure of $K$ and $K^{ab}$ is the maximal abelian extension of $K$. I want to prove the following relation $$G(\overline ...
1
vote
1answer
55 views

Galois group isomorphic to $\mathbb Z$

Does exist an example of a Galois extension $L/K$ such that $\text{Gal}(L/K)\cong \mathbb Z$? Thank you.
2
votes
1answer
33 views

Trace of an algebraic number

Let $\alpha$ have minimal polynomial $m(x) = x^n + a_{n-1} x^{n-1} + \dots + a_0.$ Show that \begin{equation*} \mathrm{Tr}(\alpha^k) = -k a_{n-k} - \sum_{i=1}^{k-1} a_{n-i} ...
0
votes
0answers
28 views

Adjoining a separable element to a field makes the extension separable

Let $k$ be a field, and $K$ be an extension. Suppose $a\in K$ is separable over $k$. What's a clever way to show that $k(a)$ is separable (i.e., that all elements of $k(a)$ have separable minimal ...
4
votes
1answer
72 views

Norm on “tower” of fields (the question comes from algebraic number theory)

Consider the field extensions $L\supset K'\supset K$, where both $L|K$ and $K'|K$ are finite and Galois. I want to prove that $$N_{L|K}(L^\ast)\subseteq N_{L|K'}(L^{\ast})$$ Maybe it is very easy, ...
1
vote
2answers
34 views

Find the fixed field of the following subgroup?

I am trying to understand some concepts via random exercises I found from past papers but this particular one, I am not sure even where to start. There aren't any solutions for the paper so would ...