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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Rational Function Field over characteristic 0

Let $K=F(x)$ be the rational function field over a field $F$ of characteristic 0, let $L_1=F(x^2)$, and $L_2=F(x^2+x)$. How to show that $L_1\cap L_2 = F$?
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How to determine what group a Galois group is isomorphic to

Consider $x^{4}-2=(x+\sqrt[4]{2})(x-\sqrt[4]{2})(x+i\sqrt[4]{2})(x-i\sqrt[4]{2}) \in \mathbb{Q}[x]$. Let $K=\mathbb{Q}(\sqrt[4]{2},i)$ be the splitting field of $x^{4}-2$. Since $K$ is a splitting ...
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A local computation of the Galois group of a polynomial

The following theorem was proved by van der Waerden in his book on algebra. His proof used a factorization of a multivariate polynomial. A paper on the computations of Galois groups said it could be ...
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1answer
978 views

Galois closure of a $p$-extension is also a $p$-extension

I'm working on a problem in Dummit & Foote and I'm quite stumped. The problem reads: Let $p$ be a prime and let $F$ be a field. Let $K$ be a Galois extension of $F$ whose Galois group is a ...
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Product of $(1-\zeta^k)$ where $\zeta$ is a $p$th root of unity

Let $\zeta$ be a $p$th root of unity over $\mathbb Q$. I am trying to show that $\prod_{k=1} ^{p-1} (1- \zeta^k) = p$. So far, all I've been able to do is show that the product is rational, since ...
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Algebraic closure of $\mathbf{Z}/p\mathbf{Z}$

Let $K$ be the algebraic closure of $\mathbf{Z}/p\mathbf{Z}$,and let $w$ belongs to $Gal(K/(\mathbf{Z}/p\mathbf{Z}))$ be the Frobenius map sending $a$ to $a^p$.I have shown that $w$ has infinite ...
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1answer
47 views

Something special for general finite extension with top field being algebraically closed

Let $K/F$ be a finite extension with $K$ algebraically closed. How can I show that $\mathrm{char}(F)=0$ and $K=F((-1)^{1/2})$ ?
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When is $X^n-a$ is irreducible over F?

Let $F$ be a field, let $\omega$ be a primitive $n$th root of unity in an algebraic closure of $F$. If $a$ in $F$ is not an $m$th power in $F(\omega)$ for any $m\gt 1$ that divides $n$, how to show ...
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Cyclotomic extension over Q special case of Kronecker Weber theorem.

For $d\in\mathbb{Q}$, how could one show that $Q(\sqrt{d})$ lies in a cyclotomic extension of $\mathbb{Q}$ without using the Kronecker-Weber theorem?
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Are absolute Galois groups compact topological groups

Let $T$ be a finite set of primes and let $K$ be the maximal extension of $\mathbf{Q}$ unramified outside $T$. We have three Galois groups: $G_{\mathbf{Q}} = ...
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Every finite group is isomorphic to some Galois Group for some finite normal extension of some field.

I'm trying to write a proof, but I know this is incorrect. Can anyone point me in the write direction? Proposition: Every finite group is isomorphic to some Galois Group $\text{Aut}_F(K)$ for some ...
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1answer
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Infinite trancendental degree.

Let $K$ be algebraically closed field, over $F$ (base field). Assume the trancendental degree of $K$ over $F$ is infinite. Please give an example of a $F$-homomorphism $K\to K$ which is not ...
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1answer
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A complete picture of the lattice of subfields for a cyclotomic extension over $\mathbb{Q}$.

Is there a good general purpose algorithm (batch of theorems) allowing one to determine the intermediate fields between $\mathbb{Q}(\zeta)$ and $\mathbb{Q}$, where $\zeta$ is some primitive root of ...
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Every finite abelian group is the Galois group of of some finite extension of the rationals

I'm trying to prove that every finite abelian group is the Galois group of of some finite extension of the rationals. I think I'm almost there. Given a finite abelian group $G$, I have constructed ...
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1answer
119 views

Finding generators of cubic Kummer extensions

Let $K$ be a number field containing $\mu_3$, the third roots of unity. Consider a monic irreducible cubic polynomial $f \in K[x]$ whose discriminant $\Delta$ is a square in $K$. Thus the splitting ...
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1answer
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How can a subfield of an abelian extension fail to be cyclic when subjected to a norm-like condition. (How can I understand the supplied explanation)

I recently posted a question on MathOverflow (if you're interested it can be found here). While some answers were quickly produced there were a few points that I found confusing. I requested some ...
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Separable extensions

I've completed an introductory course in Galois Theory, but feel my understanding of separability is poor. I think my confusions boil down to the following question: What is the relationship ...
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0answers
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Could someone give a proof of $\sum_{i=1}^{12}\sin(2^{13i+1}\pi/169)=0$?

An alternative description is: Suppose $\zeta$ is a primitive 169-th root. Prove that $$t=\sum_{i=1}^{12}\zeta^{2^{13i}}\in\mathbb{R}$$ Besides, is it possible to find out the minimal polynomials ...
3
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1answer
114 views

Cyclotomic polynomials in $K[X]$ are defined over the prime subfield of $K$

I'm having trouble with the following Lemma: $\Phi_m$ is defined over the prime subfield of $K$ (that is, over $\mathbb Q$ or $\mathbb F_p$). When $\mathrm{char}K = 0$, $\Phi_m$ is defined over ...
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1answer
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What's an example of an unsolvable finite group without an alternating group in its composition series?

The only examples I have seen have had an alternating group in the composition series.
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On solvable quintics and septics

Here is a nice sufficient (but not necessary) condition on whether a quintic is solvable in radicals or not. Given, $x^5+10cx^3+10dx^2+5ex+f = 0\tag{1}$ If there is an ordering of its roots such ...
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1answer
154 views

Difference between two statements about Galois theory

I am having trouble understanding the difference between a proposition and theorem in chapter 14 (Galois theory) of Dummit and Foote. Prop. 5 (p.562). Let $E$ be a splitting field over $F$ of the ...
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what is the simplest example of an etale cover which is not Galois?

By "Galois morphism" I mean a morphism $f: Y \to X$ such that $Y \times_X Y$ is a disjoint union of schemes isomorphic to $Y$. Let $X$ be reduced curve over afield of char. 0. I wonder what is a ...
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1answer
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Algebraic extension and infinite polynomial ring

Let $K$ a field and $F=K(\alpha_i : i\in I)$ an algebraic extension of $K$. Is it true that for all $z\in F$ there exists $i_1,\dots,i_k\in I$ such that $z\in K(\alpha_{i_1},\dots,\alpha_{i_n})$ ? ...
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2answers
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Size of Galois Group based on roots and minimal polynomial

If we know the minimal polynomial of a field extension, how can we determine the number of elements in the Galois Group? For example, take $\mathbb{Q}(\sqrt{2},\sqrt{3})$, with minimal polynomial $ ...
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1answer
273 views

Why is Klein's quartic curve not hyperelliptic

Let $X$ be Klein's quartic curve given by $x^3y + y^3z+z^3x=0$ in $\mathbf{P}^2$. It is isomorphic to $X(7)$. How do I easily show that $X$ is not hyperelliptic? I can see that $X$ is of genus $3$ ...
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2answers
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Is it true that if $K/F$ is a splitting field and $\phi \colon K \to K'$ is an isomorphism fixing $F$ then $K = K'$?

Let $K$ and $K'$ be finite extensions over a field $F$ such that $K$ is a splitting field for a polynomial $p(x)$ over $F$, and let $\varphi \colon K \to K'$ be an isomorphism which fixes $F$. ...
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Cyclic extension and norm

I am stuck by a problem in Jacobson I for a long time. It seems not too difficult. Hope anyone can help me out. It is on page 300. Let $E$ be a cyclic extension of dimension $n$ over $F$ and $\eta$ ...
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2answers
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Splitting Field over a Field

I'm having a conceptual issue. I know that a splitting field K of p(x) is the smallest field containing both Q and all the roots of p(x) What about when you aren't given a polynomial? For example: ...
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finding a fixed field of a rational function field

let G be the subset of $\operatorname{aut}_{K} K(x)$ consisting of the three automorphisms $$ x \mapsto x $$ $$ x \mapsto 1/(1-x)$$ $$ x \mapsto (x-1)/x$$ then G is a subgroup of ...
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1answer
289 views

Proof subextension of Abelian extension is also abelian

This is a property listed on MathWorld: One nice property of an Abelian extension $K$ of a field $F$ is that any intermediate subfield $E$, with $F \subset E \subset K$, must be a Galois ...
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3answers
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Inseparable, irreducible polynomials

The standard examples of irreducible, inseparable polynomials that one encounters in an introductory course on field theory all seem to have only a single root in an algebraic closure. Are there ...
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1answer
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The field of rational functions in $n$ variables is a Galois extension of the field of symmetric rational polynomials in $n$ variables.

I've been doing a little bit of field theory for number fields but not much with function fields. The question originally asked says "For some field F, show that the field $F(u_1,\ldots, u_n)$ is a ...
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1answer
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Realizing $S_n$ as a Galois group

My question is about the realization of the symmetric group $S_n$ as a galois group of a real and normal field extension $K/\mathbb Q$. As I read, such a field $K$ can be obtained as the splitting ...
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1answer
395 views

Algebraic closure of finite field

Let $N$ be an algebraic closure of a finite field $F$. How to prove that any automorphism in $\operatorname{Gal}(N/F)$ is of infinite order? I have shown: Letting $|F|=q$, 1) $N$ is the union of all ...
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1answer
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A characterisation of quadratic extensions contained in cyclic extensions of degree 4

Let $D$ be a squarefree integer. I am trying to prove that $\mathbb Q[\sqrt D]$ is contained in a Galois extension of $\mathbb Q$ with Galois group $\mathbb Z/4$ if and only if $D$ is the sum of two ...
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1answer
420 views

Cyclotomic extensions over $\mathbb{Q}$

Let $q(n)$ denote the primitive $n$th roots of unity and let $K=\mathbb{Q}(q(n))$ be the associated cyclotomic field. Let $a$ denote the trace of $q(n)$ from $K$ to $\mathbb{Q}.$ How to prove ...
2
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1answer
193 views

Extensions over finite fields

Let $K$ and $L$ be extensions of a finite field $F$ of degrees $n$ and $m$,respectively. How do I show that $KL$ has degree $\mathrm{lcm}(m,n)$ over $F$ and $K\cap F$ has degree $\gcd(m,n)$ over $F$? ...
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1answer
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Galois group of finite extensions

Given a finite extension $E/k$ of a field $k$, how do I prove the following? $$|\operatorname{Gal}(E/k)| \text{ divides } [E:k].$$ Thanks.
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1answer
100 views

Why is the trace map from a finite extension of $\mathbb{Q}_p$ continuous?

Let $k$ be a finite extension of $\mathbb{Q}_{p}$. Why is $tr_{k/\mathbb{Q}_{p}}$ a continuous map from $k$ onto $\mathbb{Q}_{p}$?
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1answer
213 views

Polynomial interpolation of the residues of a rational function

Let $g(z) = a\prod_{i=1}^N (z-\lambda_i) \in \mathbb{Q}[z]$ be square-free. At each root $\lambda_i \in \mathbb{C}$, let $r_i$ denote the residue $\mathrm{Res}_{\lambda_i} 1/g(z)$. Let $I_g(z)$ ...
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Let K/F be a finite extension, given a polynomial in K[x] find another so that their product is in F[x]

Let $K$ be a finite extension of a field $F$, and let $f(x)$ be in $K[x]$. Prove that there is a nonzero polynomial $g(x)$ in $K[x]$ such that $f(x)g(x)$ is in $F[x]$. Should I do this by induction ...
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Corestriction in Galois cohomology (Serre, Corps Locaux)

I have a question about Chapter XIV of "Corps Locaux" by Serre. Sketch of the situation: let $K$ be a field with separable closure $\overline{K}$. Let $\Gamma_K$ be the absolute Galois group. The ...
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Generator and char. polynomial for a binary Galois Field produced by an external-XOR LFSR

My question is regarding LFSRs (Linear Feedback Shift Registers), and the binary Galois Field produced by them (also commonly termed GF($2^n$) ). I understand that a given n-bit LFSR produces a ...
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1answer
476 views

Reference request: Abel or Ruffini's proof of the Abel-Ruffini theorem

The standard proof of the Abel-Ruffini theorem that people learn is based on Galois theory and the notion of a solvable group, but my understanding is that the original proof predates Galois theory. ...
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2answers
99 views

If $L/k$ is Galois and $k\subseteq K$ is any field extension, is $LK$ Galois over $L$

There is a theorem in Lang which says that if $L/k$ is Galois and $k\subseteq K$ is any field extension, if $L,K$ are subfields of a larger field, then $LK$ is Galois over $K$. I was wondering if $LK$ ...
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1answer
200 views

How to show that a subfield of a Galois extension with Galois group $S_n$ has only trivial automorphisms.

Let $K$ be the splitting field of $f(x)\in\mathbb{Q}[x]$ over $\mathbb{Q}$, of degree $n$, and suppose that $\operatorname{Gal}(K/\mathbb{Q})=S_n$. It is easy to show that his implies that $f$ is ...
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How to show that a prime degree separable field extension containing a nontrivial conjugate of a primitive element is Galois and cyclic

Let $F$ be a field and let $E/F$ be a separable extension, with $[E:F]=p$, a prime. Given a primitive $\alpha_1\in E$ with $F(\alpha_1)=E$. Let $\alpha_2\neq \alpha_1$ denote one of the $p$ conjugates ...
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2answers
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Is there something like Cardano's method for a SOLVABLE quintic.

So there is no quadratic formula equivalent for a GENERAL fifth degree equation, but is there an equivalent formula for a SOLVABLE fifth degree equation.
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471 views

If a polynomial has only rational roots does that automatically mean it is solvable?

Note that I am talking about rational roots not rational coefficients. I know that Galois theory can tell you but I want to know if knowing whether all the roots of a polynomial are rational can also ...