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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What does it mean to represent elements of an ideal?

Say I have the polynomial $x^9 + 1$ Then: $x^9 + 1 = (x+1)(x^2 + x + 1)(x^6 + x^3 + 1)$ is a complete factorization over $GF(2)$ of $x^9 + 1$ The dimension of each ideal is: length $n - ...
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Element of $\text{Aut}_{\mathbb Q}(\mathbb Q(\sqrt[3]2),\mathbb Q^{alg})$.

I have to show that the extension $\mathbb Q(\sqrt[3]2)/\mathbb Q$ is not a Galois extension by showing that $$\text{Aut}(\mathbb Q(\sqrt[3]2))\neq \text{Hom}_{\mathbb Q}(\mathbb Q(\sqrt[3]2),\mathbb ...
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Let $ L \supseteq K $ be a finite, separable extension and $ N $ its normal closure.

Prove that there is only a finite number of intermediate fields in the extension $ N \supseteq K$. I know that if $ N = K(\alpha)$ , ie, $N$ is simple, then the result holds. So I want to prove that's ...
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1answer
28 views

In what way is a quadratic extention to a finite field isomorphic to a finite field of higher order?

I have read (I don't remember where) that a finite field that is quadratically extended, say $\mathbb F_p[\sqrt 3]$ for example, is isomorphic to the finite field $\mathbb F_{p^2}$ (assuming the ...
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1answer
27 views

Separable and inseparable extensions.

I'm very confused with separable extensions. I need to prove: Let $E/F$ finite extension. Suppose there is an element $\alpha \in E$ which is not separable over $F$. Prove the existence of an ...
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1answer
32 views

Construct generator matrix given generator polynomial?

How would I take a generator polynomial and construct a generator matrix out of it for a cyclic code? For example, I have a cyclic code in: $R_{15}=GF(2)[x] / \langle x^{15} + 1\rangle$ This is ...
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Finding all elements in GF(2^4) in terms of given polynomial

I'm working with polynomials over finite fields at the moment and a question. I found this table http://www.csee.umbc.edu/~lomonaco/f97/442/Peterson_Table.html and I picked a polynomial of degree 4 ...
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If the group of automorfisms of E/F fixes exactly F, is the extension Galois?

I'm reading the wikipedia article on Galois extensions. Something confuses me a bit: A result by Artin is mentioned: if G is a finite group of automorfisms of F and the field fixed by G is E then ...
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1answer
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Normal extension, why $\mathbb Q(\sqrt[3]2)/\mathbb Q$ is not normal and $\mathbb Q(\sqrt[3]2,\zeta_3)/\mathbb Q$ is normal.

Normal extension, why $\mathbb Q(\sqrt[3]2)/\mathbb Q$ is not normal and $\mathbb Q(\sqrt[3]2,\zeta_3)/\mathbb Q$ is normal where $\zeta_3=e^{\frac{2i\pi}{3}}$. My definition of normal extension is ...
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1answer
21 views

Is the Galois group of the polynomial $x^3-3x+3$ trivial over $\mathbb{Q}(i\sqrt{15})$?

I would appreciate it if you could verify my reasoning is correct or inform me of where the flaws in my reasoning are for the following problem: "Find the Galois Group of $x^3-3x+3$ over ...
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3answers
83 views

Prove $\mathbb{Q}(\sqrt2)$ and $\mathbb{Q}(\sqrt3)$ are not isomorphic. [closed]

The title is all there is to the question. I'm not looking for a solution to the problem; my question is, why wouldn't the two be isomorphic in the first place? They are both finite dimensional vector ...
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0answers
30 views

Galois group of characteristic polynomial

Let $A$ be a square matrix with rational entries. Let $p$ be the characteristic polynomial of $A$. Let $G$ be the Galois group of $p$ over $\mathbb{Q}$. Just out of curiosity: Can anything general ...
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1answer
24 views

If $f(x) \in F[x]$ is irreducible over $F$, and let $E$ the splitting of $f(x)$.

Then, supposing that $[E:F] = p^k$, for $p$ prime. Then $f$ is solvable by radicals. I can't think in a tower of roots to solve the problem. Help, please.
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1answer
11 views

Determining Invariant Elements under a Subgroup of the Galois Group

I am currently reading J.P. Escofier's "Galois Theory" and in the text he discusses the galois group of $\mathbb{Q}(\sqrt[3]{2}, j)$ which is isomorphic to $S_3$. I have become lost in his discussion ...
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1answer
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Show that $|\operatorname{Hom}_K(L,K^{alg})|\leq [L:K]$.

Let $K$ a field and $L/K$ a field extension. I denote $$\operatorname{Hom}_K(L,K^{alg})=\{\varphi: L\to K^{alg}\mid \varphi\text{ is a field homomorphism s.t. }\varphi|_K=id_K\}$$ and $K^{alg}$ the ...
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Give an example of a finite extension of fields that is neither separable nor normal.

The title says it all. This is not homework. Haven't made much progress - I know that with $K=F_2(t)$, $L=K(\sqrt t)$ (i.e. adjoining the square root of $t$ to the function field with coefficients in ...
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63 views

Show that every finite extension of $F$ is cyclic.

Let $k$ be a field, $k^a$ an algebraic closure, and $\sigma$ be an automorphism of $k^a$ leaving $k$ fixed. Let $F$ be the fixed field of $\sigma$. Show that every finite extension of $F$ is cyclic. ...
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2answers
27 views

Linear independence over an extension

I'm working with the extension $k(p):k(p^n)$ where p is an indeterminate. I'm trying to show that ${1,p,p^2,...,p^{n-1}}$ is linearly independent on $k(p^n)$ I started by assuming they're linearly ...
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32 views

Existence of isomorphism of splitting field extending an embedding which permutes roots

Let $K/F$ be a finite Galois extension and $a\in K$. Let's say $m_a \in F[x]$ is a minimal polynomial of $a$ and $L$ is a splitting field of $m_a$ over $F$. As $F(a) /F$ is separable, $m_a$ is also ...
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Surjectivity of tensor product map

This lemma can be found in "Noether's Problem in Galois Theory" by Richard Swan at this link: ...
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42 views

show that $K/k$ is galois

Let $K$ be a finite separable extension of a field $k$ of prime degree $p$. Let $\theta$ in K be such that $K = k(\theta)$ and $\theta_1 ..., \theta_p$ be the conjugates of $\theta$ over $k$ in some ...
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1answer
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Identify the fixed field of each of these subgroups.

Let $\zeta_0=exp \left(\frac{2\pi i}{5} \right)$. Note that $\zeta_0, \zeta_0^2, \zeta_0^3, \zeta_0^4$ are the four roots of the irreducible polynomial $P(x)=1+x+x^2+x^3+x^4 \in \mathbb{Q}[x]$. Find ...
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Determining whether $Q(i)(^4\sqrt 2) :Q(i)$ is a normal extension

I'm trying to determine whether $Q(i)(^4\sqrt 2) :Q(i)$ is a normal extension I have the polynomial $x^4 -2 \in Q(i)$ Clearly all of its roots lie in $Q(i)(^4\sqrt 2)$ So we have a splitting ...
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1answer
48 views

Trying to find splitting fields over Q of $x^{19} -1$

I'm trying to find subfields L of C which are splitting fields over Q For $x^{19}-1$ I've found the roots, but since you can't express them in exact form I don't see what to do next.
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Is there a formula which gives the solutions of a general fifth grade equation by using trascendental functions? [duplicate]

Thanks to Galois we know that the solutions of a general fifth order equations cannot be expressed by means of radicals. Can we exclude that a formula exists if we are allowed to also use (any ...
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37 views

Why is the image of a $\pmod p$ Galois representation finite?

Let $\overline{\mathbb{F}_q}$ be the algebraic closure of the finite field on $q=p^r$ elements, and $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ the absolute Galois group with the profinite ...
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1answer
35 views

Is there a genus-one curve over $\mathbb{Q}$ with no points over any solvable extension?

Is there a (non-singular) genus-one curve $E$ over $\mathbb{Q}$ that is known to have no points over any solvable extension?
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$E/k$ is not seperable but still $E$ can be genreated by one element over $k$

Let $E$ be a finite extension of $k$ and let $p^r = [E:k]_{i}$. We assume that the characteristic is $p > 0$. Assume that there is no exponent $p^s$ with $s <r$ such that $E^{p^s}k$ is separable ...
2
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1answer
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Galois groups of a certain class of polynomials

I have been working on a problem in Galois Theory: We consider the polynomial f(X)=a+bX+X^n \in Q[X]. For certain coefficients a and b (in fact, an infinite number) we have $f$ irreducible and the ...
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3answers
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Find root of polynomial over finite field

Let $\mathbb{k} = \mathbb{F}_2[\alpha]$, where $\alpha$ is a root of $x^4+x+1$. I'm stuck with finding roots of $x^2 + x + 1$ in $\mathbb{k}$. I'd be greateful for any advice.
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1answer
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infinitely many Abelian extensions of $K$ whose Galois group is $G$.

Let $K$ be a finite extension of $\mathbb{Q}$ and $G$ be a finite Abelian group. Show that there exist in finitely many Abelian extensions of $K$ whose Galois group is $G$. This a exercise from Lang ...
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2answers
19 views

Proof Verification Regarding Image of $K$-homomorphisms of a Normal Extension

Let $N$ be a normal extension of $K$, where $N,K\subseteq \mathbb{C}$, and suppose that $[N:K]=n$. Show that every $K$-homomorphism $\sigma:N\to \mathbb{C}$ satisfies $\sigma (N)=N$. My Proof: Since ...
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1answer
28 views

Embdding of $\mathbb{Q}(i)$

How to show that $\mathbb{Q}(i)$ can not be embedded into a cyclic extension whose degree over $Q$ is divisible by $4$ ? Is it true ? Although I'm stuck but I've really no idea how to do it.
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Shows that some surds cannot satisfy this equation

Let $\alpha=\sqrt[3] 2$, $\beta=\sqrt[4] 5$. I would hope to show that $c_0+c_1\alpha+c_2\alpha^2=0$ is impossible, where $c_i$ are elements in the field $\mathbb{Q}(\beta)$, and $c_2$ is nonzero. ...
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“Independence” of surds

I am pondering a question on field extensions, how do we show that $\sqrt[3] 2$ is not an element of $\mathbb{Q}(\sqrt[4] 5)$? Intuitively it is "obvious" that no matter how we perform the usual ...
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1answer
45 views

If $Gal(G_f)=S_n$ then show that $Aut(\alpha)$ is trivial

Suppose $f(X) \in \mathbb{Q}[X]$ be a polynomial of degree $n$. Let $K$ be splitting field of $f$ over $\mathbb{Q}$. Suppose $Gal(K/\mathbb{Q})=S_n$ with $n>2$. Suppose $\alpha$ is a root of $f$ in ...
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An exercise related to Krull topology - showing that two bases define the same topology

The following is a definition from my lecture notes: Let $L/K$ be a Galois extension with $G=Gal(L/K)$ then the family subgroups $Gal(L/L_{i})$, where $L_{i}$ runs over all finite ...
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2answers
30 views

Irreducible implies minimal polynomial?

In the context of field theory, let's say we are finding the minimal polynomial of the $\sqrt[3] 2$ over $\mathbb{Q}$. Clearly a candidate will be $x^3-2$, which is irreducible by Eisenstein's ...
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1answer
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Formula for sum of three roots in base field

Let $\mathbb K$ be a perfect field and let $P$ be an irreducible, monic polynomial with coefficients in $\mathbb K$ : $P=X^d+\sum_{k=0}^{d-1}a_kX^{k}$. Let $\alpha_1,\alpha_2,\ldots,\alpha_d$ be the ...
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Subgroups of Index $2$ of $(\mathbb{Z}_{2})^{\aleph_{0}}$

I am studying infinite Galois theory and I proved that if $$ L=\mathbb{Q}(\sqrt{p}:\,\text{$p$ is prime )} $$ That is $L$ is the composition of all fields of the form $$\mathbb{Q}(\sqrt{p})$$ where ...
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1answer
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Equation for a circle in in cartesian produced 2D configuration space and group theory

How does the equation for a unit circle: $$ 1 = x^2 + y^2 $$ Unit Circle Relate to group theory and symmetries? It is clearly symmetric, but how is this expressed in set/group theory in terms of ...
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Constructing semidirect product out of finite fields and Galois groups and the permutation groups they induce

Let $r$ be a prime and let $K$ be a finite field of order $2^r$. Let $A$ denote the addtive group of $K$, let $M$ denote the multiplicative group of $K$ and let $H$ denote the Galois group of $K$ ...
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Minimal polynomial of $\theta^2$

I'm doing this Galois Theory question: $f = x^3 + x + 3$ is known to be irreducible and has just one real root, call it $\theta$. What is the minimal polynomial for $\theta^2$? Is it just $f$?
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Counterexample: If $f: \mathbb{Q}(\alpha) \to \mathbb{Q}(\alpha)$ is an isomorphism of fields, then $\beta=f(\alpha)$. [closed]

State whether the statement below is true or false: If $f: \mathbb{Q}(\alpha) \to \mathbb{Q}(\beta)$ is an isomorphism of fields, then $\beta=f(\alpha)$. If true, provide a proof; if false, provide ...
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1answer
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Let $K/F$ be an arbitrary extension and $a \in K$.Then is there a $\sigma \in Gal(K/F)$ such that $\sigma(a) = b$?

Let $K/F$ be an arbitrary extension and $a \in K$. If $b$ is any root of $min(F,a)$ in $K$, then is there a $\sigma \in Gal(K/F)$ such that $\sigma(a) = b$ ? That is here I want to know that the ...
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1answer
34 views

If every polynomial of odd degree is reducible in $\mathbb{F}$ then $[E:\mathbb{F}]$ is not odd

$\mathbb{F}$ is a field of characteristic zero, and every odd-degree polynomial over $F$ has a root in $\mathbb{F}$. now I have two questions:- If $E$ is a finite extension of $F$ show that ...
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36 views

Let $K=\Bbb Q(X)$ where $X=\{\sqrt p: p$ is prime$\}$. How to conclude that $|Gal(K/\Bbb Q)|>[K:\Bbb Q]$

Let $K=\Bbb Q(X)$ where $X=\{\sqrt p: p$ is prime$\}$. Then $K$ is galois over $\Bbb Q$. If $\sigma \in Gal(K/ \Bbb Q)$, let $Y_{\sigma}=\{\sqrt p: \sigma(\sqrt p)=-\sqrt p\}$. Then how to prove a) ...
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0answers
15 views

Prove that there are infinitely many intermediate field of $K=k(x,y)$ over $F=k(x^p,y^p)$.

Prove that there are infinitely many intermediate field of $K=k(x,y)$ over $F=k(x^p,y^p)$. I know $[K:F]=p^2$ and my intution is the intermediate fields are of the form $L=F(a)$ for some $a \in ...
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1answer
35 views

Algebraic Extension (Infinite)

If $E$ is an algebraic extension of the infinite field $F$, show that $|E|=|F|$. Any suggestions?
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1answer
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Proving that $Gal(K^{\text{sep}}/K)=Aut_{K}(\tilde{K})$

My lecturer told my class to understand the following equality $$ Gal(K^{\text{sep}}/K)=Aut_{K}(\tilde{K}) $$ Where $K$ is a field, $K^{\text{sep}}$ is the separable closure within an algebraic ...