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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Embedding ordered number fields into $\Bbb R$

Consider a number field $K$ equipped with a total order $\le$. Is there always a field homomorphism $\phi:K \rightarrow \Bbb R$ which respects the total order, i. e. with $\phi(x)>\phi(y)$ for ...
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Normal field extension separable over its fixed field

Let us have a field $K\supseteq E$ and $G$ be its group of automorphisms over $E$. Let the fixed field of $G$ be $K^G$. I would like to show that $K$ is separable over $K^G$. I know that for ...
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411 views

Proof that a finite separable extension has only finite many intermediate fields

Let $E/F$ be finite separable extension. Is there any proof of the fact that there are only finitely many intermediate fields without using primitive element theorem or fundamental theorem of Galois ...
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2answers
261 views

Two different definitions of separable polynomial

This is from A field guide to Algebra by Antoine Chambert Loir. A polynomial $P \in K[X]$ is separable if and only if its roots are in an algebraic closure of $K$ are simple. Here is ...
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Application of Hilbert 90 for Finite Fields

Let $k = \mathbb{F}_{p^n} = \mathbb{F}_q$ finite field of $q = p^n$ and $[K:k]=2$ Galois extension of degree 2. Then $K = \mathbb{F}_{q^2} = \mathbb{F}_{(p^n)^2} = \mathbb{F}_{p^{2n}}$. It is ...
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Field of characteristic 0 such that every finite extension is cyclic

I am trying to construct a field $F$ of characteristic 0 such that every finite extension of $F$ is cyclic. I think that I have an idea as to what $F$ should be, but I am not sure how to complete the ...
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1answer
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Finding degree of the extension [duplicate]

Is it true that the degree of extension $\mathbb Q(\sqrt {2},\sqrt {3},\sqrt {5},\dotsc,\sqrt {p_n}) / \mathbb Q$ is $2^n$ where $p_n$ is the $n$th prime number. If so, how to prove this? My idea is ...
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250 views

Necessarily complex analytic proofs in algebra.

Does anyone know of an example where complex analysis is necessary to prove something in algebra? I would be particularly interested in results from group theory or Galois theory. In an ideal ...
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1answer
61 views

Norm -1 in the extension $\,E[i]/E\,$ , where $\,E=\Bbb Q(\zeta)\,,\,\, \zeta^5 = 1$

Denote by $\zeta = \exp(2\pi i/5)$ the primitive root of unit of order 5 ($\zeta^5=1, \zeta \ne 1$). Let $E = \mathbb{Q}[\zeta]$. Then $i = \sqrt{-1} \notin E$. Let $L = E[i]$. We want to show that ...
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Field extension of composite degree has a non-trivial sub-extension

Let $E/F$ be an extension of fields with $[E:F]$ composite (not prime). Must there be a field $L$ contained between $E$ and $F$ which is not equal to either $E$ or $F$? To prove this is true, it ...
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Finding the degree of a field extension over the rationals

Let $\alpha = \sqrt{\sqrt{2}+\root 4 \of 2}$, $\beta =\sqrt{\sqrt{2}- \root 4 \of 2}$, $\gamma = \sqrt{-\sqrt{2}+i\root 4 \of 2}$ and $\delta = \overline{\gamma}=\sqrt{-\sqrt{2}-i\root 4 \of 2}$. Let ...
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closed-form expression for roots of a polynomial

It is often said colloquially that the roots of a general polynomial of degree $5$ or higher have "no closed-form formula," but the Abel-Ruffini theorem only proves nonexistence of algebraic ...
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1answer
114 views

Is $\mathbb Q\left(\sqrt{5+\sqrt{5}}, \sqrt{5-\sqrt{5}}\right)$ a simple extension?

I know that it should be because is finite and separable ($\rm{char}(\mathbb Q)=0$). However I'm having some trouble in finding the primitive element. First of all, given the irreducible polynomial ...
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Intersection of Cyclotomic Fields

How would I prove that $\mathbb{Q_m} \cap \mathbb{Q_n} = \mathbb{Q_{(m, n)}}$ (here $\mathbb{Q_n}$ denotes the $n$th cyclotomic field)? I already know of a solution involving the fact that given two ...
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356 views

why are subextensions of Galois extensions also Galois?

An algebraic extension of fields $L|K$ is defined to be a Galois extension iff the set of elements of $L$ invariant under the action of $Aut_K L$ is $K$. Apparently in the sequence of field ...
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1answer
547 views

Inverse Galois problem for small groups

I am looking for a list of all small groups (maybe order $\leq 20$) realized as the Galois groups of a polynomial over $\mathbb{Q}$, with proof. Any idea where I could find these? Partial answers or ...
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Exist automorphism from an element to its conjugate

I was asked to prove the galois group of a given normal extension is non-abelian. My original solution was to use isomorphism extension theorem but that was not taught in class. So, in my new attempt ...
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1answer
999 views

Existence of irreducible polynomial of arbitrary degree over finite field without use of primitive element theorem?

Suppose $F_{p^k}$ is a finite field. If $F_{p^{nk}}$ is some extension field, then the primitive element theorem tells us that $F_{p^{nk}}=F_{p^k}(\alpha)$ for some $\alpha$, whose minimal polynomial ...
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2answers
412 views

Intermediate fields of splitting field

I'm trying to find the intermediate fields of the extension $\mathbb Q\big /\mathbb Q(\alpha)$, where $\alpha = \sqrt{7+\sqrt{13}}$. To do so I've tried to use the Galois correspondence. I've already ...
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1answer
524 views

Finding degree of a splitting field

I am trying to find the degree of the splitting field of the irreducible polynomial $f(x)=x^4-14x^2+36\in\mathbb Q[x]$. The roots of $f$ are $\left\{\pm\sqrt{7\pm\sqrt{13}}\right\}$, so the splitting ...
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1answer
203 views

Galois group of a degree 8 polynomial

Can $D_{8}$ be the Galois group of $f$ where $f$ is a degree 8 polynomial? I was thinking that the Galois group must act transitively on the roots of $f$ (if $f$ is irreducible). In that case ...
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317 views

Finding the Galois group of $\mathbb Q (\sqrt 5 +\sqrt 7) \big/ \mathbb Q$

I know that this extension has degree $4$. Thus, the Galois group is embedded in $S_4$. I know that the groups of order $4$ are $\mathbb Z_4$ and $V_4$, but both can be embedded in $S_4$. So, since I ...
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Isomorphism with cyclotomic extension

Let $K$ be a field with $\mbox{char}(K)=0$. I know that if $\xi$ is a primitive $n$-th root of the unity and $K(\xi)\big/K$ is a cyclotomic extension, then $\mbox{Gal}\left(K(\xi)\big/K\right)$ is ...
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Quintic polynomial with Galois Group $A_5$

A recent question asks what makes degree 5 special when considering the roots of polynomials with integer coefficients etc. One answer is that the Galois Group of $S_5$ is not solvable. What I am ...
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1answer
77 views

Galois correspondence between normal groups and normal extensions

I do know that, given an extension $F\big/K$, if there is a normal intermediate extension, the corresponding subgroup of $\mbox{Gal}\left(F\big/K\right)$ is normal. The problem is that I don't see why ...
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Galois group acts transitively

The question I am dealing with is: Let F be a field, $f(x)\in F[x]$ be irreducible and let $N/F$ be normal field extension. Let $$f(x) = g_1(x) \cdot \dots \cdot g_r (x)$$ be the factorization of ...
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1answer
287 views

Doesn't Abhyankar's lemma contradict faithful flat descent (no, but I'm confused)

Let $O_K$ be a dvr with fraction field $K$. Let $L/K$ be a tamely ramified finite Galois extension. Then, Abhyankar's Lemma implies that there exists a finite Galois extension $K^\prime/K$ such that ...
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1answer
140 views

Is a finite field extension of a imperfect field imperfect

Let $K$ be a imperfect field. Let $L/K$ be a finite field extension. Is $L$ imperfect? Suppose that $L/K$ is separable. Is $L$ imperfect? Suppose that $L/K$ is Galois. Is $L$ imperfect? I'm ...
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1answer
371 views

degree of the extension galois

I have a problem with the solution of the tasks of abstract algebra.Help please. $F=({\bf Q};+;\cdot),K=({\bf R},+;\cdot)$. Determine the degree of the extension $F_K^* (\sqrt 2,\sqrt 3):F]$. ...
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230 views

Trace and Norm of a separable extension.

If $L | K$ is a separable extension and $\sigma : L \rightarrow \bar K$ varies over the different $K$-embeddings of $L$ into an algebraic closure $\bar K$ of $K$, then how to prove that ...
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1answer
385 views

Square roots of integers and cyclotomic fields

For every $ N \in \mathbb Z$ there exists an integer $n$ such that $ \sqrt N \in \mathbb Q(\zeta_n)$. I am struggling where to start this question, please suggest me few hints.
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Extensions of degree two are Galois Extensions.

This question from Allan Clark's "Elements of Abstract Algebra" Show that an extension of degree 2 is Galois except possibly when the characteristic is 2. What is the case when the characteristic is ...
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1answer
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Galois Group of Polynomial

I would like to compute the Galois group of the Polynomial $f(x)=x^5-5x^4 +10 x^3 - 10 x^2 - 135 x + 131\in\mathbb{Q}[x] $ I already know that it is irreducible in $\mathbb{Q}[x]$ via Eisenstein's ...
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1answer
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Isomorphism of Galois groups induces an isomorphism on decomposition groups?

This is a follow up question to this. Say we have Galois extensions $L ,M$ of $\Bbb{Q}$ with $L \cap M = \Bbb{Q}$ and consider the composite $K = LM$. I want to prove that $$\begin{array}{cccccc} f : ...
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1answer
151 views

Do we have such a direct product decomposition of Galois groups?

Let $L = \Bbb{Q}(\zeta_m)$ where we write $m = p^k n$ with $(p,n) = 1$. Let $p$ be a prime of $\Bbb{Z}$ and $P$ any prime of $\mathcal{O}_L$ lying over $p$. Notation: We write $I = I(P|p)$ to denote ...
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1answer
87 views

Cardano's Formulas help

I am working on solving this cubic: $x^3 +x^2 - 2 = 0$ using Cardano's explicit formulas: $$ A = \sqrt[3]{{-27\over 2}q + {3 \over 2} \sqrt{-3D}} \qquad B = \sqrt[3]{{-27\over 2}q - {3 \over 2} ...
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Adjoining a square root of the discriminant of an irreducible cubic.

Here is an exercise. Let $\delta$ be a square root of the discriminant of $P$, an irreducible cubic polynomial over a field $K$ with characteristic not equal to 2. Show that $P$ is irreducible ...
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1answer
476 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 ...
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The Galois group of $f(t)=t^3-x_1t^2+x_2t-x_3$

I'm trying to solve this question: Let $F[x_1, x_2,x_3]$ be a polynomial ring in $x_1, x_2,x_3$ over a field $F$. Let $K=F(x_1,x_2,x_3)$ be the field of rational functions (i.e., the ...
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3answers
212 views

Existence of Galois group of order 8 as $\mathbb Z_2\times \mathbb Z_4$

I'm trying to find a Galois group of $\mathbb Z_2\times \mathbb Z_4$ but, first, I'd like to know if it does exist. I know that a Galois group of order $8$ must be a subgroup of the symmetric group of ...
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1answer
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Steps missing in the proof $[L:K]=n!\ \Longrightarrow \ f\text{ is irreducible}$

Consider the following THEOREM Let $f\in K[X]$ have degree $n$ and splitting field $L/K$. Then we have $$[L:K]=n!\ \Longrightarrow \ f\text{ is irreducible}$$ and its Proof $\ $ Suppose ...
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Separability by surjectivity of Frobenius Endomorphism

I'm trying to prove the following statement: Let $F$ be a finite field of prime characteristic $p$ and let $E$ be the field generated by $F$ and the elements $\{t^{1/p},n \geq 1\}$, where $t$ ...
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1answer
142 views

Unconventional Galois theory in Qual

This seems to be an uncommon question that one may see in Qual Algebra exams, and I don't think if it was a wise choice to put this kind of questions in Qual. I consider it as a hard Galois theory ...
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$[L:K]=n!\ \Longrightarrow \ f$ is irreducible and $\text{Gal}(L/K)\cong S_n.$

How do I go about proving the following theorem ? Let $f\in K[T]$ have degree $n$ and splitting field $L/K$. Then we have $$ [L:K]=n!\ \Longrightarrow \ f\text{ is irreducible and Gal}(L/K)\cong ...
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327 views

Irreducibility in finite field implies irreducibility in $\mathbb{Z}$?

Let $p$ be a prime, $f$ be a polynomial of $\mathbb{Z}[x]$. Suppose that $f$ is irreducible in $\mathbb{F}_p[x]$. My question is : Is $f$ irreducible in $\mathbb{Z}[x]$ ? This question is ...
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Base of a Galois extension

For a field $k$, I know that $k(x_1,\cdots,x_n)/k(s_1,\cdots,s_n)$ is a finite Galois extension with Galois group $S_n$ where $s_i$ is an elementary symmetric polynomial. Thus its dimension is $n!$. ...
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166 views

Question on an example from Galois Theory- J.S.Milne

My question seems to be quite subtle, but let is see an example first. This example is appeared originally in the notes of J.S.Milne : Consider $X^5-X-1$. Modulo 2, this factor as ...
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1answer
54 views

Listing all the subfield and corresponding subgroup

Here is an exercise in the book of Dummit Foote : Find the Galois group of the splitting field of $(x^2-2)(x^2-3)(x^2-5)$ over $\mathbb{Q}$. Then list all the subgroups and the corresponding ...
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Efficient way to find Galois group

In the book Abstract Algebra of Dummit and Foote, there is a problem as follows : Let $K=\mathbb{Q}(\sqrt[8]{2},i), F_1=\mathbb{Q}(i), F_2=\mathbb{Q}(\sqrt{2}), F_3=\mathbb{Q}(\sqrt{-2})$. Prove ...
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Doubts in the fundamental theorem of algebra using Galois theory

I'm studying the Bhattacharya's algebra book, and I have the following doubts: 1-What this $g(x)$ has to do with $f(x)$? I mean why proving the splitting field of $g(x)$ over $\mathbb R$ is $\mathbb ...