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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Given $F \subset L \subset K$ where $K$ is a Galois ext. of $F$, find an example where $F \subset L$ is not a Galois ext.

I have already shown that if $F\subset K$ is a Galois extension, then for any intermediate field $L$, we have $L\subset K$ is a Galois extension. I then want to show that it's not necessarily true ...
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$\mathbb{Q}(\zeta_p)$ contains only one subfield $L$ such that $[L : \mathbb{Q}]=3$

Suppose $p$ is a prime number, $p\equiv1$ mod $3$ and $\mathbb{Q}(\zeta_p)$ is the $p$-th cyclotomic extension. Prove that $\mathbb{Q}(\zeta_p)$ contains only one subfield $L$ such that $[L : \...
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Is there a proof that this polynomial solvable by radicals?

If $f(x)$ is the minimal polynomial of $t$, a constructible number, over $\mathbb{Q}$, then is $f(x)$ solvable by radicals? It seems to be true, at least with the examples I came up with. Can it be ...
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Requesting material on Galois theory [duplicate]

Hello I wish to study Galois theory independently. I have no previous exposure to Galois theory specifically and was wondering if anyone had any tips or recommendations of good material I can use to ...
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minimal polynomial of $\alpha - \beta$ from the minimal polynomial of $\alpha$

If $f \in k[x]$ is the minimal polynomial of $\alpha \in K$, show how to write down the minimal polynomial of $\alpha - \beta$ for $\beta \in k$. I've been playing around with the minimal polynomial ...
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Minimal polynomial using Galois theory

I have a couple of questions, given below, about the following problem from a course in Galois Theory. Let $K=\mathbb{Q}(\zeta_{13})$. $K$ contains a unique subfield $L_4$ such that $[L_4 : \mathbb{Q}...
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Generator of $Gal(K/\mathbb{Q})$

Let $K=\mathbb{Q}(\zeta_5)$. Prove that there is a $\tau \in G$ such that $\tau \zeta_5=\zeta_5^2$ is a generator of $Gal(K/\mathbb{Q})$ I belive we must consider $\mathbb{Z_5}$, but I am not ...
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Galois subfields and subgroups

Find the minimal Galois extension $L$ of $\mathbb{Q}$ containing $\mathbb{Q}(\sqrt[4]{5})$ $L=\mathbb{Q}(\sqrt[4]{5}, i)$ is a splitting field of $X^4-5$ over $\mathbb{Q}$ Describe the structure ...
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I don't understand this argument about a certain Galois group.

So I'm working with $\alpha = \sqrt{5+\sqrt{5}}$ and $E=\mathbb{Q}(\alpha)$. The minimal polynomial of $\alpha$ over $\mathbb{Q}$ is $f(x) = x^4 -10x^2 +20$ and I've determined that $E$ is its ...
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Compute the Galois group of $p(x)=x^4+ax^3+bx^2+cx+d$

Compute the Galois group of the following polynomial: $$p(x)=x^4+ax^3+bx^2+cx+d$$ Step 1: Calculate cubic resolvent Step 2: Calculate the discrimimant $\gamma$ of the cubic resolvent Step 3: If $...
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Prove that $\mathbb{Q}$ has extensions of any finite degree in $\mathbb{C}$

This is a question from a course in Galois Theory and I am quite confused. In general, the degree of a field extension $E/F$ is the dimension of the vector space $E$. What would $E$ and $F$ be in ...
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Is $Q(2^{\frac14},\sqrt7)/Q(\sqrt2)$ normal?

I think it is because $(x^2-\sqrt2)(x^2-7)$ is a polynomial over $Q(\sqrt2)$ and $Q(2^{\frac14},\sqrt7)$ is the splitting field of this over $Q(\sqrt2)$ $\iff$ $Q(2^{\frac14},\sqrt7)$ is normal. So ...
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I want to show how many intermediate fields there are between $GF(3^{12})$ and $GF(3^4)$.

So $E=GF(3^{12})$ is an extension of $F=GF(3^4)$, and I wish to know how many intermediate fields $F \subsetneq K \subsetneq E$ there are. By a result in Escofier's Galois Theory I have that $G={\rm ...
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$G$ has normal subgroup of order 5

Let $L$ be the splitting field of $x^5-7$ over $Q$ and let $G=\text{Gal}(L/Q)$ (I) Prove that $G$ has a normal subgroup of order $5$ (II) Prove that $G$ has a subgroup of order $4$ that is not ...
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Find $[\mathbb Q(\sqrt3, \sqrt{5},2^{\frac13}):Q]$

$[Q(\sqrt3, \sqrt5:Q]=4$ $$ \begin{matrix} & & \mathbb Q(\sqrt3, \sqrt{5})(2^{\frac13}) & & \\ & \stackrel{a}{\diagup} & & \stackrel{b}{\diagdown} \\ \mathbb Q(\sqrt3, \...
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Prove that $L_f \vert E$ has degree $2$, or there are exactly three fields satisfying those conditions

Let $f \in \mathbb{Q}[t]$ be an irreducible polynomial of degree $3$ and $L_f \subset \mathbb{C}$ a splitting field of $f$ over $\mathbb{Q}$. Prove that there is no field $E$ with $\mathbb{Q} \subset ...
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Galois correspondence for the field extension $\mathbb{Q}(\omega_7)$

Let $E = \mathbb{Q}(\omega_7)$, where $\omega_7$ is the 7th root of unity. We know that $$\mathbb{Q}(\omega_7) \cong \mathbb{Z}_7^{\times},$$ where $\mathbb{Z}_7$ is the multiplicative group of units, ...
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$S_4/H \simeq S_3$ where $H$ is a normal subgroup

Prove that the group of permutations of four symbols $S_4$ contains a normal subgroup H such that the quotient group $S_4/H$ is isomorphic to the group of permutations of three symbols $S_3$. $...
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Use Galois Theory to prove the existence of $A$ and $B$ such that $\mathbb{Q}(\sqrt{6+3\sqrt{3}})=\mathbb{Q}(\sqrt{A}, \sqrt{B})$

Use Galois Theory to prove the existence of $A$ and $B$ such that $\mathbb{Q}(\sqrt{6+3\sqrt{3}})=\mathbb{Q}(\sqrt{A}, \sqrt{B})$ So $\mathbb{Q}(\sqrt{6+3\sqrt{3}})$ is the field of rational numbers ...
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Regarding the unique subfields of a cyclotomic field extension

What are the unique subfields $L$ of $\mathbb{Q}(\zeta_{13})$ such that $[L:\mathbb{Q}]=2, 3, 4$, and $6$? I get that the Euler's totient function of $13$ is $12$ and that must have some relation, as ...
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What does $ K^{\alpha}$ mean?

This is in context of a statement in galois theory: If $F \subseteq K \subseteq L$ and $K$ is splitting over $F$, then $K^{\alpha}=K$ for each $\alpha \in Aut(L)$.
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Given the splitting field of a polynomial, how can I show that there are three intermediate extensions which aren't normal?

So $f = x^3 +9x -2$, and $E$ is its splitting field. I need to show that there are exactly three intermediate field extensions $K$ such that $\mathbb{Q} \subset K$ is not normal. By Descartes' rule ...
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Galois Group Solution Check

Find the Galois group $G = \text{Gal}(\mathbb{Q}(\omega_{12})/\mathbb{Q})$, and its lattice of subgroups, where $\omega_{12}$ is the 12th root of unity. We have that $$\text{Gal}(\mathbb{Q}(\omega_{...
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Is $Q(5^{1/4},√11,i)/Q $ normal

I think all roots of $x^4-5$ and $x^2-11$ and $x^2+1$ are in the field but it seems impossible to find a irreducible polynomial that contains all those roots. How can we check if it is normal? I ...
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Is $Q( 5^{1/3},√7)/Q$ a normal extension?

Can someone give me a working out of this please. I don't really have any detailed examples in my notes so i have no idea about this normal extension stuff. I do know that an extension $K\subseteq L$ ...
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Finding subfields of degree $3$

Let $F$ be the splitting field of the polynomial $X^3−5$ over $Q$. Let $G = Gal(F/Q)$. (i) Determine $G$ up to isomorphism. (ii) Find all subfields $M$ of $F$ such that $Q⊆M$ and $[M :Q] = 3$. [Give ...
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Find $[Q(w,√3) :Q] $ and find a basis

Let $w∈\mathbb C$ be a root of the polynomial $X^4−12$. Determine $[Q(w,√3) :Q]$ and give a basis of $Q(w,√3)$ over $Q$. [You may express the elements of your basis in terms of w. Note that the exact ...
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Is $\mathbb Q(\sqrt{4+i\sqrt{20}},\sqrt{4-i\sqrt{20}})=\mathbb Q (\sqrt{4+i\sqrt{20}})$?

I don't know if this is true but it is trivial that the right is contained in the left. With the other inclusion, I think the only non trivial thing to check is if $A=\sqrt{4-i\sqrt{20}} \in \mathbb Q ...
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Determine $\text{Gal}(L/Q)$ and its action on a basis of $L$.

(a) Find the minimal polynomial $f$ of $√5+i$ over $Q$. (b) Find the splitting field $L$ of this $f$. (c) Determine $\text{Gal}(L/Q)$ and its action on a basis of $L$. Stuck on part c. For (a) it ...
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Galois Theory.Subgroups of Galois Group

How to List the subgroups of the Galois group in general.Im not interested in a specific Example but to make it easier. Supposes the galois group $G=Gal[Q(v,i):Q]$ $$v= \sqrt[4]{2}$$ I know how to ...
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Find the minimal polynomial $f$ of $√5+i$ over $\mathbb Q$

The candidate is $x^4-8x^2+36=0$ but we cant use Eisenstein here to prove irreducibility. What do we do?
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How is $[Q(\sqrt2, \sqrt3 ) : Q(\sqrt2)]=2$?

$\mathbb{Q}$ is the rationals. I know that $\sqrt3 \notin \mathbb{Q}(\sqrt2)$ but so what? The answer to this question seems to be based upon that. Really don't understand what that means in finding ...
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Galois group of extension generated by all cubic roots

Let $K/\mathbb{Q}$ be generated by all cubic roots of rational numbers, that is $K=\mathbb{Q}(\{\sqrt[3]{a}:a\in\mathbb{Q}\})$. I would like to understand its Galois group. I only could prove that $...
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Subgroups of $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$

If I'm understanding the main theorem of (infinite) Galois theory correctly, applied to $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, it gives us: a) all its open subgroups are $\mathrm{Gal}(\...
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How do I find the intermediate field extensions $F \subset K \subset E$ when ${\rm Gal}(E/F)$ is known?

I have the polynomial $x^3 -2 \in \mathbb{Q}[x]$, of which the splitting field is $E = \mathbb{Q}(2^{1/3}, \omega)$ where $\omega = e^{2\pi i/3}$. The Galois group of $E$ over $\mathbb{Q}$ is ...
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Bounding the degree of an algebraic extension containing solutions to polynomials

Let $F$ be a field, and let $f_{1},\ldots, f_{s}$ be polynomials in $F[x_{1},\ldots, x_{t}]$. Assume that the degree of the polynomials is bounded by $d$, by which I mean, if $m$ is any term in $f_{i}$...
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Determine $[F(a):F]$ if $a\in K$ has $k$ distinct images under Galois group

Suppose that $K/F$ is Galois extension and $a \in K$ has exactly $k$ distinct images under $Gal(K/F)$. Show that $[F(a):F]=k$. My guess is that the images of $a$ form a basis of $F(a)$ over $F$. But ...
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Use Galois theory to find all complex roots of $T^4-2T^2-\sqrt{6}T+\frac{3}{4}$

I am currently studying Galois theory and a question that often comes up is "find all complex numbers which are roots of the polynomial $T^4+aT^2+bT+c$" where the coefficients are of the form $\pm\...
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For $\alpha = \sqrt{2+\sqrt{2}}$, what is ${\rm Gal}(\mathbb{Q}(\alpha)/\mathbb{Q}$?

So $\alpha = \sqrt{2+\sqrt{2}}$, and I've already found the minimal polynomial of $\alpha$ over $\mathbb{Q}$ to be $p=x^4 - 4x^2 + 2$ and shown that $\mathbb{Q}(\alpha)$ is a normal extension. Now I ...
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Curious about field extensions and the relationship between the automorphisms of the fields' Galois groups

Let's say we have field extensions $F(\alpha)$, $F(\beta)$ and $F(\alpha, \beta)$of a field $F$. My question is this: if $\sigma \in {\rm Gal}(F(\alpha)/F)$, is it correct to say that $\sigma \in {\rm ...
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How can I solve for the roots of a polynomial that has no algebraic solution?

I've learned that polynomials of degree >= 5 (i.e. $x^5 - x -1$) are not necessarily solvable in the radicals, due to the Abel-Ruffini Theorem. My question is: given that you can't solve a polynomial ...
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What is the Galois group $Gal(E/F)$ when $F=GF(5^3)$ and $E=GF(5^{24})$

What is the Galois group $Gal(E/F)$ when $F=GF(5^3)$ and $E=GF(5^{24})$. I have attempted to describe the Galois group, but I've become stuck, and it's entirely possible that I've made mistakes as ...
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Galois Group is $\mathbb{Z}/4\mathbb{Z}$.

Let $K \subseteq L$ be a Galois field extension with Gal$(L/K) \cong \mathbb{Z}/4\mathbb{Z}$. Show that $L$ is the splitting field of a polynomial $f(x)=(x^2 −a)^2 −b$ for elements $a,b \in K$ such ...
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Galois Group of Splitting Field, $S_4$

I've shown that the polynomial $x^4+px+p \in \mathbb{Q}[x]$, where $p$ is prime, is irreducible by Eisenstein's criterion. However, it remains to be shown that the Galois group of the splitting field ...
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Galois Extension.

Suppose K is a finite field extension of $\mathbb{Q}$. Let K ⊆ L be a Galois field extension and K ⊆ K′ be a finite field extension. Show that K′ ⊆ K′L is a Galois field extension and $$\text{Gal}(K′L ...
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Intermediate field extensions of $\mathbb{Q}(\zeta_3)/\mathbb{Q}$

I'm trying to determine the intermediate field extensions of $\mathbb{Q}(\zeta_3)/\mathbb{Q}$. The minimal polynomial of $\zeta_3$ is $x^3+1$, which has roots $\zeta_3, \zeta_3^2$ and $-1$. Therefore, ...
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2answers
97 views

Why is $\mathbb{Q}(\sqrt{2})\subseteq\mathbb{Q}(\sqrt{2}+\sqrt[3]{5})\subseteq\mathbb{Q}(\sqrt{2},\sqrt[3]{5})$

In office hours yesterday my instructor said $\mathbb{Q}(\sqrt{2})\subseteq\mathbb{Q}(\sqrt{2}+\sqrt[3]{5})\subseteq\mathbb{Q}(\sqrt{2},\sqrt[3]{5})$. I know $\mathbb{Q}(\sqrt{2}+\sqrt[3]{5})\...
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3answers
46 views

Show that $E$ is the splitting field of some polynomial in $F[x].$

Let $K$ be the splitting field of some polynomial over $F$. If $E$ is a field extension contained in $K$ and $[E:F]=2,$ I want to show that $E$ is the splitting field of some polynomial in $F[x].$ I ...
0
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1answer
40 views

Will two different subgroups of a Galois group have different fixed fields?

I'm trying to figure out will two different subgroups of a Galois group have different fixed fields. Intuitively, I think they have the same fixed fields. But I am not sure. Anyone has ideas?
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1answer
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Galois Group of $\mathbb{Q}(\sqrt{ai})$

Let $a$ be a nonzero rational number and consider the extension $\mathbb{Q}(\sqrt{ai})$. My question is for what $a$ is the extension $\mathbb{Q}(\sqrt{ai})/\mathbb{Q}$ Galois, and what is the Galois ...