# Tagged Questions

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 (galois-...

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### Prove every finite extension of subfields of $\mathbb{C}$ is simple

This is a question is from a past paper, and it's only worth 8 marks, and it's really got me. I'm allowed to assume that if $L/K$ is such an extension, that there are $[L:K]$ $K$-embeddings of $L$ ...
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### Characterizing the Galois group using permutations of roots

I'm studying Galois theory at the moment. It seems to me that the fundamental motivation for the Galois group is the following. We know that if a rational function of the roots is invariant under ...
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### Why does this method of proof for $\overline{\mathbb{F}_{p}}$ not apply for $\mathbb{F}_{p}$

Let $\overline{\mathbb{F}_{p}}$ be the algebraic closure of the finite field $\mathbb{F}_{p}$. Consider $F:= \overline{\mathbb{F}_{p}}(x,y)$ and let $K:=\overline{\mathbb{F}_{p}}(x^{p},y^{p})$. Below ...
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### Splitting field of an irreducible polynomial of degree five $f \in \mathbb{Q}[X]$

Let $\Omega/\mathbb{Q}$ be the splitting field of an irreducible polynomial $f \in Q[X]$ of degree five. Show that $Gal(\Omega/\mathbb{Q})$ equals $A_5$ or $S_5$ if $Gal(\Omega/\mathbb{Q})$ has an ...
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### Identifying the Galois Group $G(\mathbb{Q}[\sqrt{2}, \sqrt[3]{2}]/\mathbb{Q})$

I am trying to determine the Galois group $G(\mathbb{Q}[\sqrt{2}, \sqrt[3]{2}]/\mathbb{Q})$. I am fairly confident I have the correct answer, but I need someone to confirm my work since I have just ...
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### Integral closures and Galois extensions

I was reading in Lang's Algebraic Number Theory (Second Edition page 15-16) and the following proposition occured. Proposition 14. Let $A$ be integrally closed in its quotient field $K$, and let $B$ ...
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### what is the minimal polynomial of $\alpha=\rho+\rho^4+\rho^{16}$ , $\rho^{21}=1$?

here is a question that I recently tried to solve: Is it possible to construct with Compass-and-straightedge the number $$\alpha=\rho+\rho^4+\rho^{16}$$ , while $\rho^{21}=1$ over $\mathbb{Q}$ ? also ...
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### Is there always an intermediate field between non-prime field extension?

If we have $[E:F]=n$, where $n$ is not a prime number but is finite, can we like prime factorize $n=p_1p_2...p_r$, so that we have $[E:F]=[E:K_1][K_1:K_2]...[K_{r-1}:F]$ and each of the $[K_i:K_{i+1}]$...
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### The Galois closure

If $\Bbb K$ is an extension of $\Bbb Q$ having degree 4, why is the Galois group corresponding to the Galois closure of $\Bbb K$ a subgroup of $S_4$?
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### What does root of unity in $\mathbb{Z}_p$ look like?

Let $p$ be an odd prime. Then by Hensel's lemma it's clear that $\mathbb{Z}_p$ contains all $p-1$th root of unity which reduces to $1$, $2$, ... , $p-1$ in $\mathbb{F}_p$. My question is do we know ...
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### find the number of solutions of the equation $a_1x_1+a_2x_2+…+a_nx_n=0$ in a linear space over Galois field

Linear space $\Bbb F_p^n$ contains $p^n$ vectors $( x_1, x_2, ..., x_n)$ with length $n$ over finite $\Bbb F_p$ Galois field comprised from $p$ elements. How many solutions in $\Bbb F_p^n$ has the ...
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### Determine $\text{Gal}(\mathbb{Q}(\alpha) / \mathbb{Q})$ where $\alpha = \omega + \omega^7 + \omega^{11}$

Suppose $\alpha = \omega + \omega^7 + \omega^{11}$, where $\omega$ is a primitive root of unity of order 19. Determine the Galois group of $\mathbb{Q}(\alpha) / \mathbb{Q}$. With which other well ...
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### Galois Theory: Finding the discriminant of a polynomial

I am very confused on how to determine the coefficient of: $f(x)=x^3-s_1x^2+s_2x-s_3$. Artin is really proving me with too much abstraction, is there an easier way to be thinking about how to do this. ...
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### Show that $\mathbb{Q}(2^{1/4}) / \mathbb{Q}$ is not Galois; prove that $\mathbb{Q}(2^{1/4}) / \mathbb{Q}(2^{1/2})$ is Galois

I would like to show that $\mathbb{Q}(2^{1/4}) / \mathbb{Q}$ is not Galois. Can I just say that it is not separable because $2^{1/4} \in \mathbb{Q}(2^{1/4})$ but its minimal polynomial in $\mathbb{Q}$ ...
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### Permutation of a fixed field is an intermediate field corresponding with the conjugate of the group corresponding to the fixed field

The following is my question: Let $K/F$ be a Galois extension with Galois group $G = Gal(K/F)$, with intermediate field $L: F \subseteq L \subseteq K$ which corresponds to subgroup $H \leq G$ by the ...
This is my question: Suppose the $K/F$ is a Galois extension with an abelian Galois group $G$. Prove that every intermediate field $L: F \subseteq L \subseteq K$ is the splitting field (over $F$) of ...
I'm dealing with this problem from the book "Field Theory" (Steven Roman) Suppose $F$ and $E$ are fields and $\sigma : F \rightarrow E$ is an embedding. Construct an extension of $F$ that is ...