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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Intermediate fields of a field extension

Let $L := \Bbb Q(\sqrt 5,\sqrt 7).$ We have proved that $L/\Bbb Q$ is galois. I have to find all the intermediate fields of $L/\Bbb Q$. So far, I have found $\Bbb Q(\sqrt 5)$, $\Bbb Q(\sqrt 7)$, ...
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Question on the proof of existence of splitting fields for a family of polynomials

I have a question regarding the following well known result: Let $C\subseteq K[x]$ be a family of polynomials. We know that $C$ possesses a splitting field over $K$. The proof I am reading goes like ...
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1answer
128 views

Beating Gauss on Irreducibility of Cyclotomic Polynomials?

I am considering how to provide an alternative proof of the lemma used in the proof that $\Phi_n$ is irreducible: Lemma: If $\Phi_n=f_1 f_2\cdots f_r$ is a factorization into monic irreducible ...
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Separability and tensor product of fields

Is it true that a finite degree field extension $L/k$ is separable if and only if $L\otimes_{k}L$ is a reduced $L$-algebra? Surely the "only if" part is true because if the extension is ...
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Let $S=\big\{\sqrt[n]{3}\colon n\in \mathbb{N}\big\}$. Is the extension $\mathbb{Q}[S]\colon\mathbb{Q}$ algebraic?

A field extension $L\colon K$ is algebraic if every element in $\alpha \in L$ is algebraic over $K$. An elemenet $\alpha \in L$ is algebraic over $K$ if there exists a polynomial $f \in K[x]$ such ...
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Systematically describing the Galois Group and Intermediate Fields

In an exercise in the textbook you are asked to describe the Galois Group and the intermediate fields of the extension $$ L=\newcommand{\Q}{\mathbb Q}\Q(\sqrt 2,\sqrt 3)\supset\Q $$ I have noted that ...
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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)}= ...
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Find a polynomial whose splitting field is $\mathbb{Q}[\alpha,i]$

Let $f(x)=x^{3}-3x+1$ and let $\alpha$ be a root in $f$. i) Show that the polynomial $f$ is irreducible in $\mathbb{Q}[x]$. ii) Show $\alpha^{2}-2$ is a root of $f$ as well, and show that all roots ...
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Permutation of roots for Galois group with six elements

We know that $\mathbb{Q}(\sqrt[3]{2},i\sqrt{3})$ is the splitting field of $x^3-2$ over $\mathbb{Q}$, and $[\mathbb{Q}(\sqrt[3]{2},i\sqrt{3}):\mathbb{Q}]=6$. Now, consider an element $\alpha$ in the ...
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Calculating polynomials in a Galois field

I'm in $\text{GF}(8) = \text{GF}(2^3)$ and have an irreducbile polynomial $p(x) = x^3 + x + 1$, then $\text{GF}(8) = \mathbb{Z}_2[x]/\langle p(x) \rangle$ . Now I want to multiply $2$ elements of ...
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Galois group command for Magma online calculator?

I need to test if a family of 7th deg and 13 deg equations are solvable. I'm new to Magma, so my apologies, but what would I type in, http://magma.maths.usyd.edu.au/calc/ to determine the Galois ...
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1answer
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Being Galois stable under completion?

Let $R$ a Dedekind Domain, $K = \mathrm{Frac}(R)$ the fraction field, $L/K$ a finite galois extension, $R'$ the integral closure of $R$ in $L$. Then $R'$ is Dedekind again. Let $\mathfrak{p} \subset ...
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Why does $\mbox{Irr}(\alpha,K)$ have distinct roots in $N$?

My textbook assumes that $N\supset K$ is normal and finite $\alpha\in N$ is separable over $K$ $N\supset K(\alpha)$ is separable and deduces (in order to prove that $N\supset K$ is separable) that ...
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1answer
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Degree of field extension question.

Last lecture it was written " $[\mathbb{Q}[X]:\mathbb{Q}]=\infty$. The elements $1,x,x^2,\ldots$ are linearly independent (but not a basis)." This confused me.. why isn't this a basis? Given any ...
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Bachelor Thesis - Galois Theory Research Topics?

I'm on the last semester of my bachelor's degree (undergrad degree) and I will be writing my thesis next semester. I have talked to a professor at my university and one of the topics he suggested was ...
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Applications of additive version of Hilbert's theorem 90

Additive version of Hilbert's theorem 90 says that whenever $k \subset F$ is cyclic Galois extension with Galois group generated by $g$, and $a$ is element of $L$ with trace 0, there exists an ...
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Integral Galois Extension (Serge Lang)

I have two questions about the proof of the following Proposition: Let $A$ be a ring, integrally closed in its quotient field $K$. Let $L$ be a finite Galois extension of $K$ with group $G$. Let $P$ ...
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A problem in field norm and division ring

let $D$ be a division ring with center $F$ and let $K$ be a maximal field of $D$. if $N_{D^*}(K*)$ be a maximal on $D^*$, then $K/F$ is galois.
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My proof: $\alpha$ is NOT separable over $K\iff\mbox{Irr}(\alpha,K)'=0$

My proof goes as follows: Proof If $\alpha$ is a multiple root of $f=\newcommand{\Irr}{\mbox{Irr}(\alpha,K)}\Irr$ then $$ \begin{align} f&=(X-\alpha)^2g\\ f'&=2(X-\alpha)g+(X-\alpha)g' ...
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My proof: Frobenius Map generates $\mbox{Gal}(\mathbb F_{p^n}/\mathbb F_p)$

I would like to ask, whether anyone can confirm or correct the following version of the proof that the Frobenius map generates the Galois Group of a finite field. Proof First note that ...
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1answer
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Is this proof that $\mathbb F_q^*$ is cyclic correct?

I would like to show that the multiplicative group of a finite field is cyclic. My goal is to use the best of my knowledge to render the argument as efficient yet simple and understandable as ...
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Understanding a Proof in Galois Theory Notes

My course lecture notes for Galois Theory make the following (standard) claim about the uniqueness of splitting fields for a given polynomial. I am struggling to understand the proof the lecturer ...
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Understanding a Proof in Galois Theory

The following is an extract from my Galois Theory course lecture notes. I understand the proof in the reverse direction so have included only the part of the proof that confuses me, even though it ...
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Can the error term involved in the PNT be expressed in a Galois theoretic framework?

According to Wikipedia, the current best error term for the prime number theorem is $\pi(x)-\mathrm{Li}(x)=O\left(x\exp\left(-\frac{A(\ln x)^{3/5}}{(\ln\ln x)^{1/5}}\right)\right)$, while RH is ...
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1answer
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Finite Fields: check my description/derivation

I am preparing for my exam in Advanced Algebra and Galois Theory, and I am trying to find an efficient way to communicate main properties of Finite Fields. If someone could check my approach and ...
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1answer
57 views

subfield maximal w.r.t. $\sqrt[3]{2}$ not in it.

Let $K$ be a field maximal with respect to the properties that $K\leq\overline{\mathbb{Q}}$ and $\sqrt[3]{2}\not\in K$(which exists by Zorn's Lemma). I was asked to show that if $L/K$ is an algebraic ...
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3answers
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Is $\mathbb Q(\sqrt2)$ the fixed field of some automorphism of $\overline{\mathbb Q}$?

Is is possible that $\mathbb{Q}(\sqrt{2})$ is the fixed field of an automorphism $\sigma$ of $\overline{\mathbb{Q}}$? Thanks in advance for your help.
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2answers
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Are groups $\operatorname{Aut}_{\mathbb{Q}}(\overline{\mathbb{Q}})$ and $\operatorname{Aut}_{\mathbb{Q}}(\mathbb{R})$ abelian?

I am tryingx to check whether the groups $\operatorname{Aut}_{\mathbb{Q}}(\overline{\mathbb{Q}})$ and $\operatorname{Aut}_{\mathbb{Q}}(\mathbb{R})$ are abelian or not. Can anyone help? Thanks!
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Galois Group of $x^5+ 5x^3 + 5x + 1$.

I've been asked to determine the Galois Group of $x^5+ 5x^3 + 5x + 1$. This is what I know so far. 1) The polynomial is irreducible. 2) Its discriminant is $78125=5^7$ Since the discriminant is ...
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1answer
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Explicit Kummer isomorphism

Let $K$ be a characteristic $0$ field containing $\mu_n$ (the $n$-th roots of unity). Then it known that the map $K^{\times} / (K^{\times})^n \to \mathrm{Hom}(G, \mu_n)$ which sends $x$ to $\sigma ...
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Compute the Galois group over $F_{101}$

The problem is as follows: Determine the Galois group of the polynomial $f(x)=x^4-2$ over the finite field with $101$ elements, $\mathbb{F}_{101}$. I am not really sure how to go about this, but ...
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Splittinf field of a product of irreducible polynomials over Finite Fields

I was wondering if someone knew a reference, that I could look up, of what I think is a fact that: If $f(x), g(x)$ are two irreducible polynomials in $\mathbb{F}_p[x]$, for $p$ a prime, of respective ...
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Express the exact root using radicals.

In Galois Theory, when a polynomial is not solvable by radicals, does that imply that the real solutions cannot be expressed exactly using the operators $+,-,\times,/$ and/or radicals?
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1answer
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Importance of continuity of Galois representations

So for a one dimensional Galois representation $\rho: G_{\Bbb Q} \to \mathbb C^{\times}$, I know that it must factor through the abelianization of $G_{\Bbb Q}$, which by the Kronecker-Weber theorem is ...
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2answers
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Galois group of $x^4-5$

How do I find the Galois group of $x^4-5$ over $\mathbb{Q}(i)$, $\mathbb{Q}(\sqrt5)$ and $\mathbb{Q}(\sqrt{-5})$? I've managed to do so over $\mathbb{Q}$ but I don't know how to find the others. I'd ...
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Representation Theory versus Galois Theory. [closed]

At my university there is a debate about whether it is better to require students to have taken a class in Galois theory or to require a class in representation theory for admission into the graduate ...
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1answer
70 views

A criterion for an extension to be Galois

This is an exercise given during my Commutative Algebra course. I reached to solve just the "if" arrow, but not the "only if". The question is: Let $F\subseteq L$ be a finite degree extension of ...
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2answers
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$X^6 + 3X^4+3X^2-1$ is the minimal polynomial of $\sqrt{ \sqrt[3]{2}-1}$ over $\mathbb Q$

It can easily be seen that $\sqrt{ \sqrt[3]{2}-1}$ is a root of $X^6 + 3X^4+3X^2-1$, which should be its minimal polynomial. Let $a=\sqrt{ \sqrt[3]{2}-1}$. Then $\sqrt[3]{2} = a^2+1$. Therefore ...
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1answer
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Galois extension of an imaginary quadratic field

This is an exercise problem from the book "Primes of the form x^2+ny^2: Fermat, Class field Theory and Complex multiplication" Question: Let $K$ be an imaginary quadratic field, and let $K\subset L$ ...
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Decomposition subgroup of Cyclic Galois Extension

My question: Say we have a cyclic Galois extension of degree $n$ over $\mathbb{Q}$. Denote the Galois group as $G$. If $H\leq G$, then does there exist a prime, $q$ in $\mathbb{Z} \subset \mathbb{Q}$ ...
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Automorphism group of an L-function

I define the notion of Galois class of L-functions as follows: $A$ is a Galois class of L-functions if and only if the following conditions simultaneously hold true: 1) $A$ is a subset of the ...
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1answer
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How to find the fixed field?

Let $\mathbb{Q}$ be the field of rational number, then the splitting field of $x^{3}-2$ over $\mathbb{Q}$ is $\mathbb{Q}[\sqrt[3]{2}, \zeta]$ where $\zeta\neq 1$ be the third root of unity. The ...
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1answer
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Representing an element of a field extension by the base field

Let $\mathbb{Q}$ be the field of rational number. Suppose that $\alpha$ is a root of an irreducible polynomial $f(x)$ of degree $n$ in $\mathbb{Q}[x]$. Then $\mathbb{Q}[\alpha]$ is the field ...
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1answer
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Show that the minimal polynomial of every element in $K=\mathbb{Q}(\zeta)$ is solvable by radicals, where $\zeta$ is a primitive 9th root of unity.

I have found the minimal polynomial if $\zeta$ over $\mathbb{Q}$ is $x^{6}+x^{3}+1$. $\mathbb{Q}(\zeta)\colon\mathbb{Q}$ is a normal and separable extension so ...
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1answer
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finding a cancelling polynomial of an algebraic number

(Note: I am not looking for the minimal polynomial) I want to find a polynomial in $\mathbb Q[X]$ which cancels $\sqrt[4]{2}+1$. I do : $$(\sqrt[4]{2}+1)^2=\sqrt{2}+2\sqrt[4]{2}+1$$ From then on, ...
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1answer
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Question regarding invariant separable elements of an algebraic closure of a field.

Let $\bar k$ be an algebraic closure of $k$ and $\alpha \in \bar k$ be separable over $k$. Suppose that $\sigma(\alpha)=\alpha$ for any non-zero ring homomorphism $\sigma:\bar k\rightarrow \bar k $. ...
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1answer
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$\operatorname{Gal}({\mathbb Q(i,\sqrt[4]{2})}/{\mathbb Q})$

I'm studying $\operatorname{Gal}({\mathbb Q(i,\sqrt[4]{2})}\Big/{\mathbb Q})$; I find it has order $8$. Any non-trivial subgroup will have order $2$ or $4$. Subgroups of order $2$ are easy to find, ...
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2answers
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Elements of order 2 in the absolute Galois group

So I remember reading once that the only element of $G=Gal(\overline{\Bbb Q} / \Bbb Q)$ that we understand is complex conjugation. Suppose we fix an embedding of $\overline{\Bbb Q}$ into $\Bbb C$. ...
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1answer
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Solving equations in cosines

Consider the equation $9x^4+27x^3-33x^2-153x-101=0$. Its Galois Group is $C_4$. That means by Kronecker-Weber theorem that it is solvable in cosines. How can you find this solution? For example, ...
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composition of field extensions

Let fields $K\subseteq L\subseteq M$. Then we know that if $L$ is a finite extension of $K$ and $M$ a finite extension of $L$, then $M$ is a finite extension of $K$. Can we generalize this property? ...