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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Why does the automorphism mapping $\omega$ to 1, not an element of Galois group

Let $L/\mathbb{Q}$ be a field extension, where $L=\mathbb{Q}(\omega)$ and $\omega=e^{\frac{2\pi i}{7}}$. In my textbook it states that $Aut(L/\mathbb{Q})=\{\sigma_i|\sigma_i(\omega)=\omega^i, 1\leq ...
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Prove that $x^p - t $ is irreducible in $\mathbb{F}_p(t)[x]$.

Prove that the polynomial $x^p - t$ is irreducible in $\mathbb{F}_p(t)[x]$. (Here $t$ is a formal variable). I know how to prove by Eisenstein's (for integral domains and ideals). However, my ...
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About the construction of resolvents in Galois theory (over $\mathbb{Q}$ in $\mathbb{C}$)

I have to say that my question is quite long and I apologize for this. The main idea is that I would like to show how to construct resolvents for any transitive subgroup of the permutation group to ...
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Why are separable and normal field extensions so called?

To my understanding: A separable extension $K/F$ is one in which the minimal polynomial of every $\alpha\in K$ has no multiple roots. A normal extension $K/F$ is one in which some polynomial $f\in ...
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Definition of Solvable by Radicals

I am currently trying to understand the definition of solvability by radicals. In the definition he takes a polynomial $f(x)$ over the field $K$ and then considers its splitting field $L$ and requires ...
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Explicitly construct a field with 729 elements

I would like to construct a field with 729 elements. I know that $729 = 3^6$, and that I have to find an irreducible monic polynomial of degree 6 over $GF(3)$. I chose the polynomial $x^6 + 2x^2 + 1$, ...
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Simplifying the Splitting field of $x^n-a$

Let $L/K$ be a field extension where $L$ is the splitting field of the polynomial $x^n-a\in K[x]$. Clearly $L=K(t,\zeta t,\ldots,\zeta^{n-1}t)$, where $t=\sqrt[n]{a}$ and $\zeta$ is the primitive ...
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If $G$ and $N$ are soluble, then $G/N$ is soluble

Let $N$ be a normal subgroup of $G$. If both $G$ and $N$ are soluble, then $G/N$ is also soluble. My attempt: $G$ is soluble, so there exists a subnormal series $1=G_0\subset ... \subset G_n=G$. I ...
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Is $A_n$ isomorphic to $C_n$ in general

Let $A_n$ denote the alternating group of order $n$, and let $C_n$ be the cyclic group of order $n$. Correct me if I'm wrong, but I know that $A_3 = \{(),(1,2,3),(1,3,2)\}\cong \{0,1,2\}=C_3$. I'm not ...
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(non Galois) correspondence

If $L/K$ is a field extension we note $\textrm{Aut}_K (L)$ the group of field automorphisms of $L$ that are fixing each element of $K$. Fix an extension $L/K$ and note $G:=\textrm{Aut}_K (L)$. If $H$ ...
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Galois group of quintic polynomial with 4 complex solution

Suppose we have an irreducible quintic polynomial $f(x)\in \mathbb{Q}[x]$ with 4 complex solutions, say for e.g. $x^5+x^4+x^3-2x^2-2x+5$. It is easy to see that the Galois group $Gal(E/\mathbb{Q})$ ...
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Galois group and solvable by radicals

I came across the following problem in an old qualifying exam which states: Show that the irreducible $h(x)\in \mathbb{Q}[x]$ is solvable by radicals if $[K:\mathbb{Q}]=25$ where $K$ is the ...
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Radical Extensions and Splitting Field

I am trying to express the splitting field of $x^4-2x^3+5x^2-2x+4$ as a radical extension of $\mathbb{Q}$. I found the roots of the above polynomial and found that the splitting field the above ...
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How to compute the fixed field of an automorphism?

Let $\sigma$ be the automophism of $\mathbb{Q}(\omega_5)$ over $\mathbb{Q}$ that sends $\omega_5$ to $\omega_5^4$. I want to find the fixed field $(\mathbb{Q}(\omega_5))^\sigma$ Since $\sigma^2$ is ...
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A shortcut in Galois theory

How could we prove Galois correspondence without using Dedekind’s Lemma on group characters, Artin’s lemma and the primitive element theorem ? I just came across Meinolf Geck's article On the ...
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Degree of C(t) as a field extension over C

Is it true that for an indeterminate t, C(t) is a field extension over C of uncountable degree? Why or why not? Thanks.
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Show that $x^n + x + 3$ is irreducible for all $n \geq 2.$

So first of all, I used the following from my lecture notes: If $f \in \mathbb{Z}[x]$ is primitive (gcd of all the coefficients is 1) - $f$ irreducible in $\mathbb{Z}[x] \Leftrightarrow$ f ...
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Clarification of absolute Galois group temrinology

In a paper by Mazur, he writes "the profinte group equal to $G_{K,S}$ for some algebraic number field $K$ and finite set of primes $S$ in $K$. I understand $G_K = \text{Aut}(\overline{K}/K)$ but not ...
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88 views

Is the sum of an algebraic and transcendental complex number transcendental?

I was wondering if the sum of an algebraic and transcendental complex number is transcendental. I was thinking if a is algebraic, and b is transcendental, then if a+b is algebraic, then a+b-a is also ...
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Conditions on cycle types for permutations to generate $S_n$

Consider the following result of Dedekind: For any polynomial $p \in \mathbb{Z}[x]$ and any prime $q$ not dividing the discriminant of $p$, if $p$ factors modulo $q$ into a product of irreducible ...
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Show that the splitting fields of $x^3 - 2$ and $x^3 - 3$ are not equal

I'm trying to solve a problem which is concerned with the size of the intersection of $H_1 = \mathbb{Q}(\sqrt[3]{2}, \zeta_3)$ and $H_2 = \mathbb{Q}(\sqrt[3]{3}, \zeta_3)$. If I can show $H_1 \not= ...
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Finite Group is the Galois Group of an extension K/F

How can I show that every finite group is the Galois group of an extension K/F where F is itself a finite extension of Q? I know the following: (1)Every finite group is contained in $S_p$ for a ...
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Polynomial in $Q[x]$ with 2 complex roots

I need to show that for any $n$, I need to show that there is an irreducible polynomial in $Q[x]$ of degree $n$ having exactly $n-2$ real roots. As a hint I have that if $f(x) \in \mathbb{R}[x]$ is a ...
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Sequence of elements $x_i \in \bar{Q}$

I am reading about applications of Galois theory to polynomials, but when using it for a degree 5 polynomial I got confused by the following: I need to show that any sequence of elements $x_i \in ...
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44 views

Show the following ring does not have nontrivial two sided nilpotent ideal

Let $k$ be a field. $A/k $ is a finite dimensional algebra with no non-trivial nilpotent two sided ideals. If $G/ k$ is Galois, then is it true that the ring $A \otimes _k G$ does not have any ...
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Every finite group is contained in $S_p$

I am reading about the inverse Galois problem. I stumbled with the problem of showing that every finite group is contained in $S_p$ for a large enough prime $p$, is this true? does anybody have a hint ...
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1answer
56 views

How to determine if a quintic polynomial is solvable by radicals

I wish to determine if $f(x)=x^5+x^4+x^3-2x^2-2x+5$ is solvable by radicals over $\mathbb{Q}$. In other words, I want to know if its Galois group is solvable. I haven't gotten anywhere trying to ...
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Matrices and Galois groups

Let $L\supseteq K$ be a finite-dimensional Galois extension with Galois group $G = \{\varphi_1,\ldots, \varphi_n\}$. Let $l_1,\ldots, l_n$ be a basis for $L$ over $K$. Prove that the matrix ...
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Galois group of $X^n-t$ over $\mathbb{C}(t)$

I would just like to check my reasoning: The polynomial $f(X)=X^n-t \in \mathbb{C}(t)[X]$ is irreducible as it is Eisenstein at the prime $t$ in $\mathbb{C}[t]$ and $\mathbb{C}(\sqrt[n]{t})$ is a ...
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Describing the Inertia group of a number field

Let $ K \subseteq L$ be number fields and $\pi$ be a prime ideal of $L$. $G = \operatorname{Gal}\left(L/K\right)$ Let $D = \{\sigma \in G\:|\:\sigma(\pi) = \pi\} $ be the decomposition group for ...
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Why restriction to $B(\alpha)$ is a homomorphism from $Gal(E/B)$ to a group with kernel $Gal(E/B(\alpha))$?

I'm reading Galois Theory for Beginners by John Stillwell. It's a good introduction, giving the essence of the idea with minimum algebra complexity. However, I'm a bit lost at his Theorem 2 (the ...
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Artin's Algebra Book Problem:Prove that splitting fields of $x^3+ex+6$ over $Q(e)$ and $x^3+\pi x+6$ over $Q(\pi)$ are isomorphic.

Assume that $\pi$ and $e$ are transcendental. Let $K$ be the splitting field of $f(x)=x^{3} + \pi x + 6$ over $F = Q(\pi)$ (a) Prove that $[K : F] = 6$. (b) Prove that $K$ is isomorphic to the ...
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Intermediate fieds for a Cyclotomic Polynomial of order $27$?

I would like to determine the Galois structure of the field $K=\Bbb Q(\zeta_{27})$--the rationals adjoined a primitive $27^{th}$ root of unity. That is to say I would like to determine the ...
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Inverse Limits in Galois Theory

I am currently taking a first course in Galois Theory and we are studying finite fields at the moment. In the lectures we have defined the inverse limit of an inverse system of finite groups and had ...
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show that $K=\Bbb F_{q^m}$, where m is the order of $q$ in the group of units $\Bbb {Z}_n^*$ of the ring $\Bbb Z_n$.

Let $q$ be power of a prime $p$, and let $n$ be a positive integer not divisible by $p$. We let $\Bbb F_q$ be the unique upto isomorphism finite field of $q$ elements. If $K$ is the splitting field of ...
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Splitting field and polynomial of minimal degree

Let's assume that we have a splitting field $F$ over $Q$ that is a finite extension. Let $p(x)$ be the polynomial in $Q[x]$ that has $F$ as a splitting field and is of minimal degree. Is it correct ...
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Given two arbitrary terms of a geometric series, can this lead to geometric series parameters that cannot be solved for in terms of radicals?

So some higher degree polynomials (and polynomial systems) can still have their roots be solved for in terms of radicals. It comes down to Galois groups and I'm a bit rusty on that subject so I don't ...
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1answer
60 views

Galois group for $x^8 - 2$

My textbook asked me to find the Galois group $G$ for $x^8 - 2$. Ok, so the roots of $x^8 - 2$ are $e^{2\pi ik/8} * 2^{1/8}:0 \le k < 8$, by my calculations, so the splitting field is ...
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Computing the minimal polynomial over a field $K$ using a finite field extension $K\subseteq E$ by means of a $K$-linear map.

I can honestly say that this site has helped me out a great deal in the past between posting questions and/or researching questions posted by others. Typically, I turn to reaching out for help if ...
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Elements of the Galois group of a polynomial acting as identity in the field $K$

For a field $K$, a polynomial $f \in K[X]$ and its splitting field $L$, we define the Galois group of the polynomial as $$\text{Gal}(f) := \text{Aut}(L/K)$$ The elements of $\text{Aut}(L/K)$ are the ...
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If K $\subset$ L, K Galois over F, L is a splitting field of f(x) over F, can we say L Galois over F

Let $F$ be a field, $f(X) \in F[X]$ an irreducible polynomial of degree $n$ over $F$, $L$ a splitting field of $f(X)$ over $F$, and $\alpha \in L$ a root of $f(X)$. If $K$ is any Galois extension of ...
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$Gal(\bar{\mathbb Q}/\mathbb Q)$ without choice, and constructive Galois theory

By this question: Algebraic closure for $\mathbb{Q}$ or $\mathbb{F}_p$ without Choice? We have that over ZF, algebraic closures of $\mathbb Q$ aren't unique. Are their Galois groups as extensions over ...
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Necessary condition for the extension $F(\alpha) /F$ to be algebraic?

Let $K/F$ be an extension of fields and let $\alpha$ be an element of $K$ such that the set $\quad X = \{ \varphi(\alpha) \; | \; \varphi \in \operatorname{Aut}(K/F) \}$ is finite. Then, if $K/F$ ...
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The fundamental theorem of Galois theory

Let $E/Q$ be a Galois extension of degree $p^2$, where $p$ is a prime number. Prove that $L/Q$ is a Galois extension for any $L \in Intermediate(E/Q)$ and find $p$ if the cardinality of ...
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Can we find the generator of the Galois group of $x^{p-1}+x^{p-2}+…1$?

$p$ is a prime. We know that $x^{p-1}+x^{p-2}+...1$ is irreducible in Q[x]. And the splitting field of $x^{p-1}+x^{p-2}+...1$ over $Q[x]$ is $Q(\xi_p)$-the primitive pth root of unity. Now I want to ...
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74 views

Concise Introduction to Galois Theory

I'm looking for a short, concise introduction to Galois Theory (but please don't assume I know anything about Galois Theory). I don't want a complete and "fat" Bourbaki-style book. My main motivation ...
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28 views

Showing that a given field is the splitting field of a given polynomial

Let $F = Z_2$; show that the splitting field of $f(x) = x^3 + x^2 + 1 \in F[x]$ is a finite field with $8$ elements. As $f$ has degree $3$, it is reducible if it has root in $F = Z_2$ but by ...
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1answer
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Let $K\subset L$ a finite field extension and $G=G(L/K)$ a Galois group of $L$ over $K$ then $|G|\leq[L:K]$.

Let $K\subset L$ a finite field extension and $G=G(L/K)$ a Galois group of $L$ over $K$ then $|G|\leq[L:K]$. I was trying to show this. By the Primitive Element Theorem $\exists\alpha\in L$ such ...
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1answer
28 views

Is it true that all the roots of the minimal polynomial of $\alpha$ are Galois Conjugates of $\alpha$?

To be more precise, Suppose [K:F] is Galois, $\forall \alpha \in K$ $\alpha \not\in F$, let $m_\alpha$ be the minimal polynomial of $\alpha$ in $F[x]$. I know that $\phi \in Gal(K:F)$, $\phi(\alpha)$ ...
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1answer
83 views

Lattice of intermediate fields of $\mathbb{Q}(i,\sqrt{2},\sqrt{3}) / \mathbb{Q}$

I need to construct the lattice of intermediate fields of $\mathbb{Q}(i,\sqrt{2},\sqrt{3}) / \mathbb{Q}$. I think I know how to do this by simply listing them, but it seems that the picture I get ...