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

2
votes
1answer
161 views

Showing that a field extension is Galois

Let $F$ be an extension field of a field $K$. Let $E$ be an intermediate field such that $E$ is Galois over $K$, $F$ is Galois over $E$, and every $\sigma\in\mathrm{Aut}_{K}E$ is extendible to $F$. ...
0
votes
0answers
58 views

Is there a way to prove that $x^n+x^m=C, x$ $\&$ $C \in ℝ, n$ $\&$ $m \in ℕ$ has no general solution for $x$?

I think I have made myself clear in the title and is hoping for someone to show this with as simple math as possible. Since I'm only 17 and haven't learned all of the complex mathematics yet. However ...
3
votes
1answer
347 views

Lagrange interpolation of Galois field functions

I'm trying to understand how to apply Lagrangian interpolation for a function $F : GF(2^k) \to GF(2^k)$, but am having some difficulty. Suppose I have a function $F : GF(2^2) \to GF(2^2)$ given by the ...
3
votes
1answer
127 views

A generalization of Galois connections

Let $f(x)\ast y \Leftrightarrow x\ast g(y)$ for some binary relation $\ast$ and functions $f$ and $g$. This is a generalization of Galois connections. Are things like this studied before? Note that ...
1
vote
1answer
125 views

Proving the order of a Galois group is equal to the dimension of $F$ over its fixed field w

Suppose $F/K$ is a finite dimensional field extension and $G = Aut_KF$. Let $G'$ be the fixed field of $F/K$, i.e. the set of members of $F$ which are fixed by every element of $G$. Before the ...
1
vote
1answer
145 views

Fixed field of automorphisms group of a non-Galois extension

If $k$ is a perfect field and $K/k$ a non-Galois extension, how can I show that the fixed field of $Aut_k(K)$ is not $k$ please ?
1
vote
1answer
48 views

Number of solutions of $P(x_1, …, x_n) = 0$ in $\mathbb{F}_q^n$

I have this exercise : $q = p^r$ (it is not clearly precised in the exercise that $r = n$). and $\mathbb{F}_q$ is the finite field of cardinal $q$. Let's $P \in \mathbb{F}_q[X_1, ..., X_n]$ of degree ...
8
votes
5answers
475 views

Prove $\sqrt[3]{3} \notin \mathbb{Q}(\sqrt[3]{2})$

I've tried solving $\sqrt[3]{3} =a + b* \sqrt[3]{2}+c* \sqrt[3]{4}$, but there is no obvious contradiction, even when taking the norms/traces of both sides. I can't think of another approach. This is ...
2
votes
2answers
108 views

No rational solutions of a system of equations

Please show that there does not exist $(a,b,c)\in\mathbb{Q}^3$ such that \begin{matrix} a^2b+2b^2c+2ac^2=0\\ a^2c+ab^2+2bc^2=0\\ a^3+2b^3+4c^3+12abc=3. \end{matrix} I'm able to show that this ...
7
votes
1answer
558 views

An exercise on cyclic extensions of Hungerford's book, Algebra.

I'm trying to show the following exercise of the book, it is in the section of cyclic extensions and says the following: Let $\overline{\mathbb Q}$ be a fixed algebraic closure of $\mathbb Q$, let ...
2
votes
1answer
174 views

Application of the Fundamental theorem of Galois Theory

This is an example from my lecture notes. I'm not sure what it is called so sorry for the poor title. As far as I can tell it is an application of the Fundamental Theorem of Galois Theory. So Let K ...
2
votes
1answer
53 views

There is always a Galois subfield

Let K be subfield of $\mathbb C$ s.t. $|K:Q|<\infty$. Prove that there is a subfield $L \subset \mathbb C$ with $K \subset L$, $|L:Q|<\infty$ and with L Galois over $\mathbb Q$. I'm also given ...
1
vote
2answers
128 views

Definitions of norms of an element in a field.

Let $K$ be a finite dimensional Galois extension field of $F$, then the norm of $u\in K$ is defined by $$N_{K/F}(u) = \sigma_1(u) \cdots \sigma_n(u),$$ where $Aut_F K = \{\sigma_1, \ldots, ...
2
votes
2answers
73 views

Special elements of fields extensions

I was wondering if there is a method to find all elements $w\in F(\alpha_1,\ldots,\alpha_n)$ such that $F(w)=F(\alpha_1,\ldots,\alpha_n)$, where $\alpha_1,\ldots,\alpha_n$ are algebraic over the field ...
-1
votes
1answer
67 views

field extension is Galois extension if and only if this extension are both normal extension and separable extension

I knew field extension $L/K$ is Galois extension if and only if this extension are both normal extension and separable extension. I want know example: i) normal extension but not Galois extension. ...
0
votes
1answer
140 views

The field of Laurent series over $\mathbb{C}$ is quasi-finite

How can I prove that the field of Laurent series over $\mathbb{C}$ is quasi-finite, which means that it has a unique extension of degree $n$ for all $n \geqslant 1$ ? The article ...
3
votes
0answers
129 views

Galois group of $x^{4}+ax^{2}+b$

Let $x^{4}+ax^{2}+b\in K[x]$ (with $\mathrm{char}K\neq 2$) be irreducible with Galois group $G$. I have shown that (a) if $b$ is a square in $K$, then $G$ is the Klein 4-group, (b) if $b$ is not a ...
2
votes
1answer
394 views

Splitting field that isn't a Galois extension

I'm trying to find a counter-example to following statement: if $K$ is the splitting field of $g\in F[x],$ then the extension $K/F$ is Galois. I know the statement is true if $g$ is separable, ...
2
votes
5answers
613 views

Minimal polynomials of $\sin(\pi/8)$ and $\cos(\pi/9)$

Is there an "easy" way to find the minimal polynomials of $\sin(\pi/8)$ and $\cos(\pi/9)$ without the help of any computer programme? If I knew $\sin(\pi/8)=\frac{\sqrt{2-\sqrt2}}{2}$ then it would ...
3
votes
1answer
109 views

Galois group of a quintic

What is the Galois group of $x^5-4x+12$? I'm able to show it has to be either the Frobenius $F_{20}$ group, or the dihedral group $D_{10}$. Is there a less computationally heavy way to determine? What ...
-1
votes
1answer
77 views

galois theory radical extension..

If p(x) is solvable by radicals over F; then prove that there is a radical tower F = B0 B1    Bt such that Bt is a splitting field of some polynomial over F (or Bt=F is a normal extension).
-2
votes
1answer
111 views

Galois theory Radical extension

Prove that any splitting field $K/F$ containing a radical extension $R_{t}/F$ is itself a radical extension.
1
vote
0answers
540 views

$\mathbb Q(\sqrt 2,\sqrt 3,\sqrt 5)=\mathbb Q(\sqrt 2+\sqrt 3+\sqrt 5)$ - Generalisation?

Problem: We know that $\mathbb Q(\sqrt 2,\sqrt 3,\sqrt 5)=\mathbb Q(\sqrt 2+\sqrt 3+\sqrt 5)$. Generalise this fact. My idea is to use the (proof of the) primitive element theorem. Looking at ...
4
votes
1answer
55 views

No field properly between $\mathbb Q$ and $E$ iff $G(K/\mathbb Q) \cong A_4$ or $S_4$

Let $f(x) \in \mathbb Q[x]$ be irreducible of degree 4. Let $\alpha$ be a root of $f(x)$. Let $E = \mathbb Q(\alpha)$ and let $K$ be the splitting field of $f(x)$ over $\mathbb Q$. Prove that there is ...
1
vote
1answer
378 views

Degree of the splitting field of $x^{p^2} -2$ over $\mathbb{Q}$, for prime p.

I've already shown that the degree of the splitting field of $x^p-2$ over $\mathbb{Q}$ is $p(p-1)$ as follows: $x^p-2$ has roots $\sqrt[p]{2}\omega_{k}$ for $k=0,1,...,p-1$, where the $\omega_{k}$ ...
5
votes
2answers
142 views

$f \in K[X]$ of degree $n$ with Galois group $S_n$, why are there no non-trivial intermediate fields of $K \subset K(a)$ with $a$ a root of $f$?

Let $K$ be a field and $f \in K[X]$ of degree $n$ with Galois group $S_n$. Let $a$ be a root of $f$, $L = K(a)$, and let $E$ be a intermediate field of the extension $K \subset L$. Prove that $E = K$ ...
2
votes
0answers
90 views

Galois group of $X^4+4X^3+12X^2+24X+24$

What is the best method to find the Galois group of $P = X^4+4X^3+12X^2+24X+24$ over $\mathbb{Q}$ ? First, I don't manage to show that $P$ is irreducible : Eisenstein doesn't work. I know that its ...
1
vote
1answer
136 views

Question about separable extension

Here is an assignment problem.$\\$ Let $K/F$ be a finite field extension and $S=\{u\in K\ |\ \sigma(u)=u\ ,\forall \sigma\in \operatorname{Gal}(K/F)\}$. Suppose $S=F$. Prove or disprove that $K/F$ ...
1
vote
0answers
226 views

Trace and cyclotomic field

Let $K=\mathbb{Q}(\zeta_p)$ be the cyclotomic field of $p$th roots of unity for the prime $p$ and let $G=\operatorname{Gal}(K/\mathbb{Q})$. Let $\zeta$ denote any $p$th root of unity. Please show that ...
4
votes
1answer
153 views

How to prove that this infinite extension of $\mathbb{Q}$ is Galois

Let $K_0=\mathbb{Q}$ and for $n>0$ define $K_{n+1}$ as the extension of $K_n$ obtained by adjoining to $K_n$ all the radicals of elements in $K_n$. Let $K$ be the union of the subfields $K_n$. ...
1
vote
1answer
48 views

Degree of the splitting field

I'm looking for a proof that, if $K/k$ is the splitting field of a polynomial $P$, and $z_1, ..., z_n$ the roots of $P$ in an algebraic closure of $k$, then $$[K : k] | \prod_{i=1}^n [k[z_i]:k]$$ Is ...
2
votes
0answers
212 views

Galois Theory References

This may not initially be a well posed question, but I'm looking for a good reference on Galois theory that covers it from the viewpoint of the symmetry in roots of an irreducible polynomial and not a ...
6
votes
2answers
151 views

Usage of algebraic geometry in understanding the total Galois group of the rational

A professor of mine remarked in class that: "The tools via which we understand the total Galois group of the rationals comes from algebraic geometry". Could anyone shed some light on this remark, or ...
1
vote
0answers
125 views

Why is the intersection of Q($\sqrt[n]{a}$) and Q(nth root of unity) Galois? (a>0, n an integer)

With this, I can show the intersection is either Q or Q($\sqrt{a}$). All I have is that their intersection must be real and all subfields of Q($\sqrt[n]{a}$) are of the form Q($\sqrt[d]{a}$) where ...
3
votes
1answer
411 views

Proving $ x^p - a $ is irreducible iff $ a $ is not a $p$-th power of an element

I have the following homework question, which I've managed to do the forward implication of, but not the other direction: Let $\mathbb{K}$ be a field, $ a\in\mathbb{K}$ and $p$ be a prime. Show that ...
2
votes
3answers
100 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, ...
0
votes
1answer
295 views

A radical extension with a non-radical subextension

For a Galois Theory class I've been asked to find a radical extension with a non-radical subextension (all over $\mathbb{Q}$). So, I'm looking at the splitting field of $x^7 - 1$, namely ...
2
votes
1answer
175 views

Show the extension is not cyclic

Let $D\in\mathbb{Z}$ be a squarefree integer and let $a\in\mathbb{Q}$ be a nonzero rational number. Please show that $\mathbb{Q}(\sqrt{a\sqrt{D}})$ cannot be a cyclic extension of degree 4 over ...
2
votes
3answers
90 views

Two quadratic fields over $\mathbb{Q}$

I'm having a bit of trouble showing that the two quadratic fields $\mathbb{Q}[X]/(X^2+1)$ and $\mathbb{Q}[X]/(X^2+3)$ over $\mathbb{Q}$ are not isomorphic (as fields). Could someone help me? Perhaps ...
1
vote
2answers
83 views

Larger Theory for root formula

Consider the quadratic equation: $$ax^2 + bx + c = 0$$ and the linear equation: $$bx + c = 0$$. We note the solution of the linear equation is $$x = -\frac{c}{b}.$$ We note the solution of the ...
2
votes
1answer
228 views

Fixed field of automorphisms determined by $t\mapsto at+b$.

Suppose $E=\mathbb{F}_p(t)$, the field of rational functions in a transcendental $t$ over the finite field of $p$ elements. Suppose $G$ is the group of field automorphisms fixing $\mathbb{F}_p$ ...
6
votes
1answer
260 views

Irreducibility over $\mathbb F_p$ - A useless hint?

Dummit and Foote, 13.5.5: For any prime $p$ and nonzero $a \in \mathbb F_p$ prove that $x^p-x+a$ is irreducible and separable over $\mathbb F_p$ The question goes on to suggest two approaches ...
0
votes
1answer
915 views

Find the Galois group of $x^3 -2$ over $\mathbb{Q}$. [duplicate]

Show that it is a non-abelian group of order 6, then under the Galois correspondence find the (fixed) subfield corresponding to the subgroup of G of order 3. I've found the splitting field which is ...
6
votes
2answers
140 views

Factoring $x^{255} -1 $ over $\Bbb F_2$

How would I factor the above polynomial in this binary field? We just completed a course in Galois Theory, and I'm stuck on how to efficiently factor this polynomial. I tried considering computing all ...
5
votes
2answers
467 views

Radical extension

Let $K=\mathbb{Q}(\sqrt[n]a)$ where $a\in\mathbb{Q}$, $a>0$ and suppose $[K:\mathbb{Q}]=n$. Let $E$ be any subfield of $K$ and let $[E:\mathbb{Q}]=d.$ Prove that $E=\mathbb{Q}(\sqrt[d]a)$. It's ...
1
vote
1answer
120 views

Definition of Galois Group

I'm revising for a module on Galois Theory and have trouble understanding the definition for a Galois group of a field extension $K:F$. Define the Galois group of $K:F$ as $\Gamma(K:F)=\{\sigma \in ...
1
vote
1answer
229 views

Fixed field of group of automorphisms on $\mathbb{C}(t)$.

I've been stuck on this problem tonight. Suppose $E=\mathbb{C}(t)$ where $t$ is transcendental over $\mathbb{C}$, and let $\omega$ be a primitive cube root of unity. Let $\sigma$ be the automorphism ...
1
vote
1answer
212 views

Generator of rational functions unchanged under $\sigma(X) = X + 1$

Let $L = K(X)$ be the field of rational functions over a field $K$ with characteristic $p > 0$, and let $\sigma \in \operatorname{Aut}_K(L)$ with $\sigma(X) = X + 1$. Show that $G = ...
4
votes
2answers
684 views

How to find the splitting field and Galois group of $x^6 -4x^3 +1$?

I am trying to find the splitting field $L$ of the $x^6 -4x^3 +1$ over $\mathbb{Q}$, and its Galois group. Here are some things I have figured out. I did the usual trick of solving for $x^3 = 2\pm ...
3
votes
2answers
126 views

Question on relation between normal subgroups and normal extensions in Fundamental Theorem of Galois Theory.

I'm self studying Jacobson's Basic Algebra I but I'm getting hung up on the proof of the Fundamental Theorem of Galois Theory in Jacobson's book on page 239. Let $G=\operatorname{Gal}(E/F)$ for a ...