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 (galois-...

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Galois extension definition.

Let $L,K$ be fields with $L/K$ a field extension. We say $L/K$ is a Galois extension if $L/K$ is normal and separable. I don't fully understand this definition, is it saying that 1) $L$ has to be ...
0
votes
1answer
32 views

interchanges/transpositions (how to read)

I have came across this before and just again now, in the same form of which I'm struggling to understand. Although I know it's link to parity, as a perm group pi: $$ \pi = \begin{pmatrix} 0 & 1 ...
11
votes
0answers
67 views

Is it possible for an irreducible polynomial with rational coefficients to have three zeros in an arithmetic progression?

Assume that $p(x)\in \Bbb{Q}[x]$ is irreducible of degree $n\ge3$. Is it possible that $p(x)$ has three distinct zeros $\alpha_1,\alpha_2,\alpha_3$ such that $\alpha_1-\alpha_2=\alpha_2-\alpha_3$? ...
5
votes
1answer
86 views

Degree of the difference of two roots

Let $f\in \mathbb Q[x]$ be an irreducible monic polynomial of degree $n$ and let $\alpha,\beta\in \overline{\mathbb Q}$ be two distinct roots of $f$. Is it possible to find a lower bound on the degree ...
4
votes
1answer
58 views

Bound for the degree

Let $K$ be a perfect field and let $f\in K[x]$ be a monic irreducible polynomial of degree $n$. Denote by $\alpha,\beta$ two distinct roots of $f$. Is the following bound true? $$ [K(\alpha-\beta):K]\...
4
votes
1answer
30 views

Cyclic Galois group of even order and the discriminant

I am stuck on the following problem: Let K be a field of characteristic $\neq 2$ and $f\in K[X]$ a separable irreducible polynomial with roots $\alpha_1,\ldots \alpha_n$ in a splitting field $...
0
votes
1answer
9 views

$L$ and $F$ are separable extension of $K$ and stay in the same bigger field $M$. Prove that $LF$ is separable extension of $K$.

$L$ and $F$ are separable extension of $K$ and stay in the same bigger field $M$. Prove that $LF$ is separable extension of $K$. I see the solution of the finite case but in the infinite I just think ...
6
votes
0answers
74 views

Generalizing Ramanujan's cube roots of cubic roots identities

(This extends this post.) Define the function, $$\sqrt[3]{G(t)} = \sqrt[3]{t+x_1}+\sqrt[3]{t+x_2}+\sqrt[3]{t+x_3}\tag1$$ where the $x_i$ are roots of the cubic, $$x^3+ax^2+bx+c=0\tag2$$ While $G(t)...
1
vote
3answers
49 views

How to evaluate GF(256) element

I wonder is there any easy way to evaluate elements of GF$(256)$: meaning that I would like to know what $\alpha^{32}$ or $\alpha^{200}$ is in polynomial form? I am assuming that the primitive ...
9
votes
2answers
188 views

What's the explanation for these (infinitely many?) Ramanujan-type identities?

Define the function, $$F(\beta) := \sqrt[3]{\beta+x_1}+\sqrt[3]{\beta+x_2}+\sqrt[3]{\beta+x_3}\tag1$$ where, $$x_1 =2\cos\big(\tfrac{2\pi }{7}\big),\;x_2 =2\cos\big(\tfrac{4\pi }{7}\big),\; x_3 = ...
1
vote
2answers
16 views

Let $K_0$ denote the prime subfield of $K$ then $\text{Gal}(K/K_0)=\text{Aut}(K)$

Let $K_0$ denote the prime subfield of $K$ then $\text{Gal}(K/K_0)=\text{Aut}(K)$. This is a statement in the book I am looking at, it is given without proof or further explanation. I don't see ...
2
votes
1answer
42 views

Galois group of function field

Let $K$ be an arbitrary field, and $K(t)$ denote the field of rational functions in $t$, i.e. function field on $K$. If $K$ is algebraically closed field, then $\mathrm{Gal}(K(t),K)\cong \mathrm{...
5
votes
2answers
73 views

When is a 5th degree polynomial with at least 1 non-real root solvable by radicals?

Let $f(X)$ be an irreducible polynomial of degree 5 with coefficents in the field of rational numbers $\mathbb{Q}$. Assume that $f$ has at least one non-real root in the complex field $\mathbb{C}$. ...
1
vote
0answers
38 views

Given some arbitrary roots of a polynomial p(x,y,z,…) with integer coefficients, is it possible to tell if p has a root in the Gaussian integers?

I'm trying to find if p(x,y,z,...)=0 has a Gaussian integer root (more specifically, I want to find if p has a Gaussian integer root where the imaginary components are even, but if that can't be done, ...
2
votes
1answer
44 views

Does there exist an algebraic solvability algorithm?

I was ruminating over quintics and got curious about the following idea. Consider a quintic equation: $$ Q(x) a_0 + a_1 x + a_2 x^2 + ... a_5 x^5$$ Such that the solutions to $$ Q(x) = 0 $$ Are ...
17
votes
5answers
1k views

Ramanujan-type trigonometric identities with cube roots, generalizing $\sqrt[3]{\cos(2\pi/7)}+\sqrt[3]{\cos(4\pi/7)}+\sqrt[3]{\cos(8\pi/7)}$

Ramanujan found the following trigonometric identity \begin{equation} \sqrt[3]{\cos\bigl(\tfrac{2\pi}7\bigr)}+ \sqrt[3]{\cos\bigl(\tfrac{4\pi}7\bigr)}+ \sqrt[3]{\cos\bigl(\tfrac{8\pi}7\bigr)}= \sqrt[3]...
1
vote
2answers
35 views

If $p(X)$ is irreducible polynomial with $\alpha$ being a root in field K then $p(X)=min(K,\alpha)$, is it right?

I think that my question is very stupid but I just wanna ask that: If $p(X)$ is monoic and irreducible polynomial with coefficient in K and $\alpha$ being a root in field $K(\alpha)$ then $p(X)=min(K,\...
4
votes
1answer
37 views

Relating the class number of a field, and of its normal closure

Suppose I take a number field $ K $, not necessarily Galois, with class number $ h_k $ (over $ \mathbb{Q} $). Write $ \overline{K} $ for the normal closure of $ K $. What, if anything, can be said ...
4
votes
2answers
60 views

Galois group of a palindromic polynomial is not $S_n$?

Let $f(x) = a_nx^n+\cdots+a_0 \in \mathbb{Q}[x]$ be a palindromic polynomial; that is, the coefficients of $f$ satisfy $a_n = a_0$, $a_{n-1} = a_1$, and more generally $a_{n-i} = a_i$ for all $0\leq i\...
0
votes
1answer
24 views

$\mathbb{Z}_p$-extensions of CM-fields

I am trying to prove some consequences of Iwasawa's Theorem for CM-fields. There is a sequence of CM-fields $$K=K_0\subseteq K_1 \subseteq \dots \subseteq K_\infty$$ so that $K_\infty/K$ is a $\mathbb{...
4
votes
1answer
40 views

L/K is Galois extension Prove that $Gal(L/N)=\cap_{\phi\in G} \phi H \phi^{-1}$

Suppose that $L/K$ is Galois extension with $G=Gal(L/K)$ and $M$ is immediate field of it. If $N\subseteq L$ is normal closure of $M/K$ and $H=Gal(L/M)$ then prove that $Gal(L/N)=\cap_{\phi\in G}{\phi ...
2
votes
4answers
53 views

Meaning of $Gal(L/L)$ for some field $L$?

In my notes it says $Gal(L/L)=1$ and I am confused on the notation clearly there is only one automorphism of $L$ that map all elements of the base field $L$ to itself namely the identity map. But what ...
4
votes
1answer
42 views

An isomorphism between the fields $\mathbb{Q}(e)$ and $\mathbb{Q}(\pi)$

How can i prove that $\mathbb{Q}(e)\cong \mathbb{Q}(\pi)$? Also, is this isomorphism a valid one? We have that $\mathbb{Q}(e)=\left \{ \cfrac{f(e)}{g(e)} \mid f(x),g(x)\in\mathbb{Q}[x], g(e)\neq0 \...
1
vote
1answer
41 views

Is every subgroup of $S_n$ the Galois group of some extension of $\mathbb{Q}$?

Is every subgroup of $S_n$ the Galois group of some extension of $\mathbb{Q}$? It is well known that most (in some suitable sense) polynomials $f \in \mathbb{Q}[x]$ of degree $d$ and coefficients $|...
35
votes
2answers
606 views

Is the Galois group associated to a random polynomial solvable with probability 0?

Choose a random polynomial $P\in\mathbb{Z}[x]$ of degree $n$ and coefficients $\leq n$ and $\geq-n$. Let $r_1,\ldots,r_n$ be the roots of $P$ and consider $$G=\operatorname{Gal}(\mathbb{Q}(r_1,\...
4
votes
0answers
30 views

Sum of divisors in $\mathbb{Z}_2[x]$

Let $A$ denote a polynomial on $\mathbb{Z}_2[x]$ and $\sigma(A)$ denote the sum of divisors on $A$. Let$$A = x^h(x + 1)^k P^l Q^{2n - 1},$$where $P$, $Q$ are irreducible polynomials with degree at ...
1
vote
1answer
62 views

Is this polynomial solvable by radicals?3

Suppose you have a field $\mathbb{F}$. Show that the polynomial $x^n-n\cdot1_{\mathbb{F}}\in \mathbb{F}[x]$, where $n\geq 2$ is solvable by radicals.
2
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0answers
31 views

Proof check: commutation of Galois automorphisms and complex conjugation in CM-fields

Let $K/\mathbb{Q}$ be a Galois CM-field with $Gal(K/\mathbb{Q})=:G$ and $J_\mathbb{C}$ be the complex conjugation. Since $K$ is a CM-field one can show, that $$J:=\phi^{-1}\circ J_\mathbb{C}\circ \phi=...
3
votes
1answer
79 views

Reference request: Galois descent

What is a classic (perhaps even original) reference for Galois descent? I know that it can be seen as a special case of faithfully flat descent (for which FGA and SGA I is the usual reference) and ...
1
vote
2answers
60 views

Aut$(K/F)$ permutes roots of polynomial.

Let Aut$(K/F)$ is the set of all automorphism from $F$ to $K$, where $K$ is a galois extension of $F$. Let $f(x) \in F[x]$ and $\alpha$ be a root of the polynomial $f(x)$. I am able to prove that for ...
13
votes
2answers
1k views

Galois group over $p$-adic numbers

Can one describe explicitly the Galois group $G=\operatorname{Gal}(\overline{\mathbb Q_p}/\mathbb Q_p)$? I only know the most basic stuff: unramified extensions of $\mathbb Q_p$ are equivalent to ...
12
votes
1answer
248 views

Is there a proper subfield $K\subset \mathbb R$ such that $[\mathbb R:K]$ is finite?

Is there a proper subfield $K\subset \mathbb R$ such that $[\mathbb R:K]$ is finite? Here $[\mathbb R:K]$ means the dimension of $\mathbb R$ as a $K$-vector space. What I have tried: If we can find ...
3
votes
3answers
79 views

Is there a non trivial normal subgroup of a group $G$, where $|G|=pm, \ \gcd(p,m)=1$?

In fact, the exercise I'm in is this: Suppose you have an irreducible polynomial $f(x)\in \mathbb{Q}[x]$ of degree $p$, where $p$ is a prime. Also suppose that $K$ is a splitting field of $f(x)$ over ...
0
votes
0answers
30 views

If $K$ is a Galois extension of $Q$, can we find $E$ so that $Gal(E/K)$ = $S_n$, for all $n$? [duplicate]

If $K$ is a Galois extension of $Q$, can we find $E$ so that $Gal(E/K)$ = $S_n$, for all $n$? edit: I would appreciate if this wasn't closed, the question that was refered to does not provide an ...
0
votes
0answers
43 views

Classification of set of algebraic integers in $F$

Let $F$ be a field with $\mathbb{Q} \subseteq F \subseteq \mathbb{C}$, where $F/\mathbb{Q}$ is a finite abelian Galois extension. Then can we classify set of algebraic integers in $F$ ?
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2answers
38 views

irreducibility of a bivariate polyonimal over a finite field

Let $\mathbb{F}_q$ be the finite field with $q$ elements. Consider the bivariate polynomial $$P(x,y)=y^2- x(x-1)(x-a)(x-b),$$ where $a\neq b$, and $a,b \neq 0,1$ are some arbitrary elements of $\...
2
votes
1answer
51 views

Possibilities for $\deg f$ if $\text{Gal}(f/\mathbb{Q})=Q_8, D_8$

Let $f$ be irreducible over $\mathbb{Q}$ with splitting field $F$. Suppose $\text{Gal}(F/\mathbb{Q})$ is either $D_8$ or $Q_8$. What are the possibilities for $\deg f$? I'm using Dummit & Foote,...
0
votes
3answers
49 views

a question about the Galois group for $x^3 - 2$

I know this question has been posed before, but there are some things that I'm still confused about. First of all, I know that the splitting field for this particular polynomial will be the field ...
3
votes
1answer
60 views

Norm of roots of unity conjugated by Galois automorphisms in CM-fields

Let $(K_n)_{n\geq0}$ be a sequence of CM-fields, so that $K_0\subset K_1\subset\dots$ with $[K_{n+1}:K_n]=p$ for all $n\geq0$. For $n\geq0$ let $W_n$ be the group of the roots of unity in $K_n$. Now ...
0
votes
0answers
45 views

Finding the Galois group of $x^4+5x^2+5$

Find the Galois group of $f(x)=x^4+5x^2+5\in \mathbb{Q}[x]$. This is solved here, Exersice 3: https://math.berkeley.edu/~serganov/114/solhwg.pdf I have a question about it (I will not write all the ...
3
votes
2answers
63 views

Degree of Splitting Field to Prove Irreducibility

Let $f(x) \in F[x]$ have degree $n>0$ and let $L$ be the splitting field of $f$ over $F$. Show that if $[L:F]=n!$ then $f(x)$ is irreducible over $F$. My approach: I attempted to prove the ...
2
votes
2answers
59 views

Find the Galois group of $x^9+x^6+x^3+1$.

Question: Let $L$ be the splitting field of $f = x^9+x^6+x^3+1 $ over $\mathbb{Q}.$ Find the Galois group $ G = \text{Gal}\left(L/\mathbb{Q}\right). $ Initially, I decomposed $f$ into its ...
2
votes
1answer
32 views

$\text{deg}(f)$ is not divisible by $[L:F]$

I am trying to recall an exam question so I am sorry if this question doesn't make full sense. I think some people would know what the actual wording should be after reading it. $F \subseteq L$ is ...
1
vote
3answers
60 views

Constructible $n$-gons

Let $\xi$ be the primitive root of unity. If $n=5$, then the minimal polynomial of $xi$ over rationals would have degree $5-1=2^2$ which is a fermat prime so $5$-gon is constructible. If $n=8$, ...
1
vote
0answers
78 views

Trigonometric solution to solvable equations

The algebraic equations in one variable, in the general case, cannot be solved by radicals. While the basic operations and root extraction applied to the coefficients of the equations of degree $ 2 $ ,...
1
vote
0answers
51 views

Question about the Fundamental Theorem of Galois Theory

This is something more like a small doubt than some problem that I need help with. I'm doing and exercise that is asking me to find subextensions of a given extension of all posible orders, and find ...
1
vote
1answer
39 views

Why are these two calculations in $GF(2^5)$ not equal [closed]

I have a quick question about Galois fields, since there seems to be something I have misunderstood way back in university. Addition and subtraction in Galois fields are both done using XOR ...
0
votes
1answer
31 views

Galois group question (involving the primitive n-th root of unity)

Suppose $\gamma$ is the primitive n-th root of unity. That is, $\gamma = e^{\frac{2 \pi i}{n}}$. I have to find the subgroup of $G(\mathbb{Q}(\gamma)/\mathbb{Q})$ that fixes the field $\mathbb{Q}(\...
2
votes
1answer
46 views

Galois Groups of Polynomials

Computing Galois groups of polynomials is a kind of standard thing to do in algebra, so I think this question goes without much further motivation. It is known that every element of the Galois group ...
3
votes
4answers
116 views

minimal polynomial $\zeta_n$ and $\zeta_n^p$ is the same for any prime $p$ not dividing $n$

I want to prove that for any prime $p$ not dividing $n$, $\zeta_n$ and $\zeta_n^p$ have the same minimal polynomial over $\mathbb{Q}$. My proposed proof, Suppose $\zeta_n$ is a primitive $n$th root ...