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 ...

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Concise Introduction to Galois Theory

I'm looking for a short, concise introduction to Galois Theory (but please don't assume I know anything about Galois Theory). I don't want a complete and "fat" Bourbaki-style book. My main motivation ...
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Showing that a given field is the splitting field of a given polynomial

Let $F = Z_2$; show that the splitting field of $f(x) = x^3 + x^2 + 1 \in F[x]$ is a finite field with $8$ elements. As $f$ has degree $3$, it is reducible if it has root in $F = Z_2$ but by ...
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Let $K\subset L$ a finite field extension and $G=G(L/K)$ a Galois group of $L$ over $K$ then $|G|\leq[L:K]$.

Let $K\subset L$ a finite field extension and $G=G(L/K)$ a Galois group of $L$ over $K$ then $|G|\leq[L:K]$. I was trying to show this. By the Primitive Element Theorem $\exists\alpha\in L$ such ...
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Is it true that all the roots of the minimal polynomial of $\alpha$ are Galois Conjugates of $\alpha$?

To be more precise, Suppose [K:F] is Galois, $\forall \alpha \in K$ $\alpha \not\in F$, let $m_\alpha$ be the minimal polynomial of $\alpha$ in $F[x]$. I know that $\phi \in Gal(K:F)$, $\phi(\alpha)$ ...
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Lattice of intermediate fields of $\mathbb{Q}(i,\sqrt{2},\sqrt{3}) / \mathbb{Q}$

I need to construct the lattice of intermediate fields of $\mathbb{Q}(i,\sqrt{2},\sqrt{3}) / \mathbb{Q}$. I think I know how to do this by simply listing them, but it seems that the picture I get ...
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If $K$ is a field of charateristic $0$ then every irriducible polynomial over $K$ is separable over $K$

Let the seguent propositions: Lemma $1$: A polynomial $f \not=0$ over a field $K$ has a multiple zero in a splitting field if and only if $f$ and $Df$ have a common factor of degree $\ge1$ ...
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Show that $\sqrt{2}\notin \mathbb{Q}(\sqrt[4]{3})$

I want to show that $\sqrt{2}\notin \mathbb{Q}(\sqrt[4]{3})$. I think it would be easier to prove it using the following: $\mathbb{Q}\subset\mathbb{Q}(\sqrt{3})\subset\mathbb{Q}(\sqrt[4]{3})$. Then ...
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Writing elements of a finitely generated field extension in terms of the generators

I have a question that arose in the process of solving a different problem in Galois theory. That problem asked to show that if $\alpha_1, \dots, \alpha_n$ are the generators of an extension $K/F$, ...
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Galois group of a polynomial over $\mathbb C(t)$

I am learning for my exam tomorrow and I am facing the following task: Compute the Galois group of $x^3-2tx+t$ over $\mathbb C(t)$ At first I want to show that this polynomial is indeed irreducible. ...
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Quintic polynomial problem without analysis

Just wondering: Is there a purely algebraic proof of the existence of an irreducible quintic polynomial over $\mathbb{Q}$ with exactly three real roots? Sure, it's easy to give concrete examples, but ...
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Possible dimension of number field

What are the possibilities for the dimension $[E:\mathbb{Q}]$ of a galois extensions $E/\mathbb{Q}$? I think I have a proof that $[E:\mathbb{Q}]$ can be any number. First note that it will be ...
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Information about the cyclic group $C_{p^n}$, where $p$ is prime and $n\geq 2$

My question is quite generic and I need it to have an idea of the knowledge of this group. In particular, I am interested in the extension of number fields whose Galois group is $C_{p^n}$ (if there is ...
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Galois group over the field of rational functions

I am looking to find the Galois group of $x^3-x+t$ over $\mathbb{C}(t)$, the field of rational functions with complex coefficients. I have shown that the automorphisms of the rational function field ...
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Finding a primitive element of $\mathbb{Q}(\alpha,\beta)$ ($\alpha^m=2$, $\beta^n=3$)

I want to prove the following fact: Let $m,n \in \mathbb{N}$ be coprime and $\alpha, \beta \in \mathbb{C}$ with $\alpha^m=2$ and $\beta^n=3$. Then $\alpha\beta$ is a primitive element of ...
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Cyclotomic fields: Determining the fixed field

Let $K=\mathbb Q$ and $L=\mathbb Q(\zeta_8)$, where $\zeta_8$ is a primitve $8th$ root of unity. I have to determine the Galois group of this extension, the subgroups of it and the associated ...
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K-monomorphism and K-automorphism in Galois Theory

I was trying to show a lemma but i had some problem: If $L:K$ is finite then every $K$-monomorphism $L→L$ is an automorphism. I was thinking: "Is it necessary that the extension is finite?" My ...
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Boolean Polynomial Mutliplication modulo an irreducible polynomial

I am currently reading a paper on the Mathematics of RAID 6 by Peter Anvin and cannot get my head around the notation or results used describing multiplication by {02} (hexadecimal). Here is how he ...
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Find the Galois group of the polynomial

Consider the polynomial $ f(X)= X^4 + 9 \in \Bbb Q [X]$. Show that $f$ is irreducible and find $Gal (K/ \Bbb Q )$, where $K$ is the splitting field of $f$. For the first one I used Eisenstein ...
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Biggest splitting field degree given a polynomial of degree n

It's a well know fact that, given $f(x) \in \mathbb{K}[x]$ with $\deg(f) = n$, and being $\mathbb{L}$ its splitting field, we have that $[\mathbb{L}:\mathbb{K}] \leq n!$ What I'd like to know are ...
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Compute Galois group and minimal polynomial

Let $ \zeta \in \Bbb C$ be a seventh root of $1$. Find the minimal polynomial of the element $ \alpha = \zeta ^{-1} + \zeta$ over $ \Bbb Q$ and show that if $K= \Bbb Q ( \alpha ) $, then $ K/ \Bbb Q$ ...
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Why is $F_5(\root{15}\of t)$ not normal over $F_5(t)$?

(I'm asking this to understand the solution of this question. My idea for a proof so far: $f:=X^{15}-t\in F_5(t)$ is the minimal polynomial of $\root{15}\of t$ in $F_5(t)$, so it suffices to show ...
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Find the monic irreducible polynomials of degree 2 in $F_3$

I know that there are 9 distinct monic polynomials of degree 2 in $F_3$. To find which are irreducible, should I just list them all out and check each one, or is there a better way of checking this?
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Cardinality of the ring $F_3[x] / (x^2-x+1)$

How would I find the number of elements of the ring $F_3[x] / (x^2-x+1)$? I know that $x^2-x+1$ is not prime/irreducible, since gcd($x^2-x+1$, $x^3-x^2-1$) = 3. Can anyone provide some tips?
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Cyclotomic fields of finite fields

There is a fact that if $K=\mathbb Q$, then the Galois group of the $n$th cyclotomic field over $\mathbb Q$ is isomorphic to $\mathbb Z_n^{*}$. In the case were $K$ is an arbitrary field, we have ...
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Inertia field of a compositum.

My question is regarding composite field extensions. The problem is motivated but not explicitly stated in the chapter 4 exercises of Marcus' Number Fields, in particular ex. 31 c) We first provide ...
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Valuation rings are conjugate

Let $F/K$ be a finite Galois extension where $(K,v)$ is a valued field (i.e. $v$ is a valuation on $K$). Let $w_1,w_2$ be extensions of $v$ to $F$. Then, we have associated valuation rings $O_{w_1}$ ...
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Can $\text{Aut}(L^s_K/K)$ and $\text{Aut}(L/K)$ be canonically identified?

I am reading a Galois theory textbook. The following is a point that I don't understand. Suppose that $L/K$ is a finite normal extension with $\text{Aut}(L/K)$. Denote by $L^s_K$ the set $\{x\in ...
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Find the degree of $[\mathbb{Q}(\mu):\mathbb{Q}(\mu + \mu ^{-1})]$

Find the degree of [$\mathbb{Q}(\mu):\mathbb{Q}(\mu + \mu ^{-1})$]. Here $\mu$ is the primitive $nth$ root of 1. I know for $n$ being a prime, [$\mathbb{Q}(\mu):\mathbb{Q}$]=n-1, but not sure about ...
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Generating elements of a Galois Field using an irreducible polynomial

I am practicing some cryptography problems and I am having problems with one involving Galois Fields and irreducible polynomials. Here is the problem: Using the irreducible polynomial $f(x) = x^5 ...
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Galois group of $x^4 + 2$ over $\mathbb{Q}$

I have a problem finding Galois group of $x^4 + 2$ over $\mathbb{Q}$. Not sure whether to start. It's irreducible over $\mathbb{Q}$ and also have 4 different roots.
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Determining of the intermediate fields of the $12$th cyclotomic field

Let $\zeta$ be a 12th primitive root over $\mathbb Q$. Determine all intermediate field of $\mathbb Q(\zeta)/\mathbb Q$. My problem is that this is a task from an old exam where you were not allowed ...
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Construction of a field with $8$ elements.

Could someone tell me if one can build a field with $8$ elements?
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Primitive roots in $\mathbb{F}_7[x]/(x^2+1)$

Are $2+x$ and $1+x$ primitive roots in $\mathbb{F}_7[x]/(x^2+1)$? I'm having trouble with the arithmetic associated with finding primitive roots.
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If to E/F, every irredicible poly over at F with a root in E has all its roots in E, then E/F is Galois?

I know the converse case is right, but I don't know this case, especially I can't find or proof the separable poly over F such that E is the splitting field.
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Subextensions of cyclotomic field

Let $p$ be a prime and $\zeta_p$ be a $p^{th}$ primitive root of unity. Let $G=\operatorname{Gal}(\mathbb{Q}(\zeta_p)/\mathbb{Q})$, it is well known that every sub-extension of $\mathbb{Q}(\zeta_p)$ ...
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image of Kummer isomorphism

Let $K$ be a finite extension of the field of $p$-adic numbers which contains the $p$-roots of unity. There is an isomorphism $K^{\times} / (K^{\times})^p \to \mathrm{Hom}(\mathrm{Gal} (\bar{K} / K), ...
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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 ...
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Is there a Galois extension over Q with degree 11?

I tried to use Q($\xi_{23}$), Q($\xi_{89}$), in the first one, it is Galois with degree 22, it does have a fixed field of degree of 11 as the group of order 22(namely G)has a 2-sylow group, but as ...
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If $Aut_K(L)$ operates transitive over the zeros of $f$, then $f$ is irreducible

Let $K$ be a field, $f\in K[X]$ a separable polynomial and $L$ a splitting field of $f$. Show that if $Aut_K(L)$ operates transitive over the zeros of f, then f is irreducible. Can someone help?
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extension field homomorphisms

Let F be a field, E a finite field extension of F, K the field of separable elements of E over F, C an algebrically closed field containing F. Is it true that every F-homomorphism from K to C extends ...
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Cyclotomic polynomials and Galois groups

According to this question I want to extend the question from there. Lets consider again the galois extension $\mathbb Q(\zeta)/\mathbb Q$ where $\zeta$ is a primitive root of the $7^{th}$ cyclotomic ...
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Galois theory in reverse

Let $K/F$ be a Galois extension. The fundamental theorem of Galois theory says that there is a bijection between subfields of $K$ containing $F$ and subgroups $H$ of $G$, where $G$ is the Galois group ...
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Cyclotomic polynomials and Galois group

Let $\zeta\in \mathbb C$ be a primitive $7^{th}$ root of unity. Show that there exists a $\sigma\in \operatorname{Gal}(\mathbb Q(\zeta)/\mathbb Q)$ such that $\sigma(\zeta)=\zeta^3$. I already know ...
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Computing $\Phi_i(X)\in\mathbb F_3[X], i=1,\ldots,6$

I want to calculate the first six cyclotomic polynomials over $\mathbb F_3$; e.g, $$ \Phi_i(X)\in\mathbb F_3[X],\; i=1,\ldots,6. $$ How is this different to computing cyclotomic polynomials over ...
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Converse of the theorem “lf a real number $c$ is constructible, then $c$ is algebraic of degree a power of 2 over the field $\mathbb{Q}$ of rationals”

In Thomas W. Hungerford's ALGEBRA book Chapter V, Proposition 1.16 (page # 239) states that lf a real number $c$ is constructible, then $c$ is algebraic of degree a power of 2 over the field ...
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Separability over intersection of intermediate fields

Let $E/F$ be a finite degree normal extension. Let $K$ and $L$ be intermediate fields such that $E$ is separable over both $K$ and $L$. Show that $E$ is separable over $K\cap L$. So I was thinking ...
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Galois comodules

I tried to figure out why Galois comodules is a generalization of several Galois aspects, but I could not? I am really interested in Comodule theory, and I am very curious to know the answer for ...
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Galoisgroup of a polynomial

Task: Determine the galois group of $x^4-4x^2-11$ over $\mathbb Q$ The roots are obviously $\pm\sqrt{2\pm\sqrt{15}}$ But I have problems in checking whether just $\sqrt{2+\sqrt{15}}$ is generating ...
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Computing the galois group of $x^3+3x+1$

I want to calculate the galois group of the polynomial $x^3+3x+1$ over $\mathbb Q$. And I am struggling in finding the roots of the polynomial. I only need a tip to start with. Not the full solution ...
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The Galois correspondence on finite extensions

One way to exhibit the bijective nature of the Galois correspondence for finite Galois extensions involves a claim of the following sort: If $L$ is a finite extension of $K$, and if $G$ is a ...