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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Galois Connection Between Posets

I have the next doubt about this problem: In a Galois Connection between posets, show that the subset $\{p\mid p=RLp\}$ of $P$ is equal $\{p\mid p=Rq \; for\; some \; q\}$ and give a bijection from ...
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“Relative unsatisfiability” of SAT instances

There's a natural way to view any SAT instance as a variety: just replace the Boolean algebra $2$ of truth values with the corresponding Boolean ring $\mathbb{Z}/2\mathbb{Z}$. (See my answer to Is ...
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Is the Galois group of a given polynomial always a subgroup of the Klein-$4$ group?

Let $f(x) = (x^2-ax+b)(x^2-cx+d)$ be a separable polynomial with rational coefficients. Is it true, that its Galois group over the rationals is always a subgroup of the Klein four group $C_2 \times ...
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Proving that isomorphism of field extension is an equivalence relation

I'm reading Stewart's Galois Theory Third Edition. In Chapter 4 he gives the definition of the isomorphism of field extension as follows: An isomorphism between two field extension $l : K ...
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Algebraically closed field of characteristic zero

I have a question about a proof which looks similar to the algebraically closedness of $\mathbb{C}$: Let $F$ be a field of characteristic $0$, such that every odd polynomial over $F$ has at least one ...
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The maximal subfield of $\mathbb C$ not containing $\sqrt2$

Related: Does a maximal subfield of $\mathbb C$ not containing $\sqrt{2}$ have index $2$? He said, "...fixed field is an extension of $K$ which doesn't contain $\sqrt{2}$, and thus must be $K$ ...
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The Galois group of automorphisms on the splitting field of the polynomial x^5 - 11.

I think that the splitting field (the smallest subfield of C that contains all the roots of x^5 - 11) is Q adjoined with r and z where r is the real solution of x^5 - 11 = 0 and z is the 5th root of ...
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Embedding fields into the complex numbers $\mathbb{C}$.

Let $k$ be a field of characteristic $0$ with $\mathrm{trdeg}_\mathbb{Q}(k)$ at most the cardinality of the continuum. I want to prove the existence of a field homomorphism $k\rightarrow\mathbb{C}$. ...
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Is a polynomial which is invariant in the roots of some separable polynomial also invariant in the usual sense?

Let $\alpha_1,\cdots,\alpha_n \in \mathbb{C}$ be the roots of a separable polynomial with rational coefficients. Let $K := \mathbb{Q}(\alpha_1,\cdots,\alpha_n)$. Then the field extension ...
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possible degrees of the minimal polynomial of t over F [closed]

Please help me with this Abstract Algebra question: Let $F \subset K$ be a Galois extension with Galois group isomorphic to the alternating group $A_4$. Let $t \in K$ be an element. Determine the ...
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Determine whether the splitting field of a polynomial contains a subfield M such that M:$\mathbb {Q}$ is not normal

For the following polynomials I need to find out if the splitting field over $\mathbb {Q}$ has a subfield M such that M:$\mathbb {Q}$ is not normal. 1) $ x^6-7$ 2)$ x^3 + 3x +3 $ 3)$x^{100} - 1$ ...
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A Galois theory sanity check about conjugates.

Here is my question... If $L/K$ is an algebraic extension and $\alpha,\beta \in L$ are $K$-conjugates (that is, they have the same minimal polynomial), is it always true that there exists some ...
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Degree of the splitting field of $ x^3-5 $ over $\mathbb{Q}$

I am trying to find the degree of the splitting field for $ x^3-5 $ over $\mathbb{Q}$. I have so far: The splitting field will be $\mathbb{Q}(\sqrt[3]{5},u)$ where u is the 3rd root of unity. So ...
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Primitive Element theorem, permutations

Let $K = \mathbb{Q}(\alpha_1,\alpha_2,...\alpha_n)$, where the $\alpha_i$ are the roots of some irreducible polynomial (and hence they are pairwaise distinct since the polynomial is separable). Then ...
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Is a finite simple extension of fields in characteristic zero already normal?

Let $k(\alpha) / k$ be a finite separable simple extension, $char(k) = 0$. Is $k(\alpha) / k$ already a normal extension? I can't come up with a counterexample or a proof that it is normal.
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Does a cubic polynomial split in linear factors over the rational numbers?

Let $\alpha + \beta + \gamma = a$ , $\alpha \beta + \alpha \gamma + \beta \gamma = b$, $\alpha \beta \gamma = c$, $\alpha^2 \beta + \gamma^2 \alpha + \beta ^2 \gamma = d$, where $\alpha, \beta, ...
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Alternating irreducibility of polynomial $t^n + t^{n-1} + \ldots + t + 1$?

I was doing a problem on Stewart's Galois Theory. In section 3, we were asked to decide the irreducibility or otherwise of the following polynomials: $t^4 + t^3 + t^2 + t + 1$ $t^5 + t^4 + ...
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The irreducibility of $t^4 + 2$ over $\mathbb{Z}_5$

I'm reading Stewart's Galois Theory Third Edition Example 3.23, where we are proving that $f(t) = t^4 + 15t^3 + 7$ over $\mathbb{Z}$ is irreducible by showing its mapping to $\mathbb{Z}_5$ is ...
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A degree $4$ polynomial whose Galois group is isomorphic to $S_4$.

I am reading an article about Galois groups. The article states that: It can also be shown that for each degree $d$ there exist polynomials whose Galois group is the fully symmetric group $S_d$. ...
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Higher ramification groups of Galois extension of order $p^2$

Let $p\in \mathbb{Z}$ be a prime number and $K/\mathbb{Q}$ be a Galois extension of degree $p^2$ over $\mathbb{Q}$. Suppose that $P\subset \mathcal{O}_K$ is the only prime ramified over $p$. Let ...
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Use the identity $\cos 3\theta = 4 \cos^3\theta- 3 \cos \theta$ to solve the cubic equation $t^3 + pt + q = 0$ when $p, q \in \mathbb{R}$.

I'm self studying Ian Stewart's Galois Theory and this is Exercise 1.8 from his Third Edition: Use the identity $\cos 3\theta = 4 \cos^3\theta- 3 \cos \theta$ to solve the cubic equation $t^3 + ...
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find an irreducible polynomial

I'm working on the following problem: Let $K/\mathbb{Q}$ be a Galois extension with $Gal(K/\mathbb{Q})$ isomorphic to $A_5$, which is the group consisting of all even permutations of 5 objects. Show ...
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Let $E/F$ a field extension. If $x\in E$ is separable over $F$ is $E/F$ separable?

Let $E/F$ a field extension. If $x\in E$ is separable over $F$ is $E/F$ separable ? I would say yes since the fact that $x$ separable over $F$ implies $E(x)/F$ separable, an since $E=E(x)$ then $E/F$ ...
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Degree of the splitting field of $X^4-3X^2+5$ over $\mathbb{Q}$

I would like to know how to solve part $ii)$ of the following problem: Let $K /\mathbb{Q}$ be a splitting field for $f(X) =X^4-3X^2+5$. i) Prove that $f(X)$ is irreducible in $\mathbb{Q}[X]$ ...
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Proof of $Gal(K_1K_2K_3/K)\cong Gal(K_1/K)\times Gal(K_2/K)\times Gal(K_3/K)$

$\newcommand{\Gal}{\text{Gal}}$ Let $K_1/K$ and $K_2/K$ two galois extension. I have a theorem that says that $K_1K_2/K$ is a galois extension and that if $K_1\cap K_2=K$ then ...
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Degree of a splitting field over finite field

Given the polynomial $f(x)=x^4+1$ in the field $\mathbb{F}_p$, prove that $[F: \mathbb{F}_p]\leq 2$, where $F$ is the splitting field of $f$. Suppose $a$ is one of the roots of $f$. Then (if ...
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Find primitive elements of a splitting field

Find all primitive elements from a splitting field of polynomial $x^3 - 5 = 0 $ over a field $Q$ and describe its Galois group. I found splitting field, but how to find which elements in that ...
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Examples where $H\ne \mathrm{Aut}(E/E^H)$

If $E/F$ is a field extension, and $H$ is a subgroup of $\mathrm{Aut}(E/F)$, it is quite trivial to see that $H\subset \mathrm{Aut}(E/E^H)$. Since the theorem only shows the inclusion relationship, ...
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Galois groups: Why does |G(E/F)|=[E:F]?

In case the notation I am using is not familiar to you, $G(E/F)$ is the Galois group of $E$ over $F$ and $[E:F]$ means the dimension of the vector space $E$ over the field $F$. Theorem : Let $f(x)$ ...
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The cubic equation $x^3 - 4 x^2 + x + 1 =0$

The cubic polynomial $P(x) =x^3 - 4 x^2 + x + 1$ has discriminant $\Delta = 169 = 13^2$ which tells us that the extension $\mathbb{Q}(a)/\mathbb{Q}$ is normal, where $a$ is any root of the equation ...
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Maximal separable sub-extensions and Galois groups.

I'm reading Neukirch's ANT book and at page 56 (bottom) it seems that the author is assuming the following fact: Let $L|K$ be a finite and normal extension of fields and consider the maximal ...
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Fixed subfield of complex conjugation

Suppose that $L \subseteq \mathbb{C}$ is the splitting field of a polynomial $f \in \mathbb{Q}[x]$, and suppose that $f$ has non-real roots. Then complex conjugation (let's call it $\sigma$) is a ...
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Splitting field of $x^3-10x+10$ over $\mathbb{Q}(\sqrt{13})$

Is it possible to find the splitting field of $x^3 - 10x + 10$ over $\mathbb{Q}(\sqrt{13})$? I think this polynomial can't be factorized.
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Under which conditions this condition of normality of groups translates into normality of extensions

Let $E/K$ be a Galois extension and $H \leq Gal(E/K)=: G$ be a subgroup. Under which hypothesis is the following true: $H$ is normal in G if and only if $E^{H}/K$ is normal. This looks similar to a ...
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On Galois groups and $\int_{-\infty}^{\infty} \frac{x^2}{x^{10} - x^9 + 5x^8 - 2x^7 + 16x^6 - 7x^5 + 20x^4 + x^3 + 12x^2 - 3x + 1}\,dx$

Given the solvable decic (among many in this database), $$P(x) = x^{10} - x^9 + 5x^8 - 2x^7 + 16x^6 - 7x^5 + 20x^4 + x^3 + 12x^2 - 3x + 1$$ we have, $$\int_{-\infty}^{\infty} \frac{x^2}{P(x)}\,dx = ...
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Show that $F_1\cap F_2/K$ is normal if $F_1/K$ and $F_2/K$ are normal.

I have to show that $F_1\cap F_2/K$ is normal if $F_1/K$ and $F_2/K$ are normal. What I know is that \begin{align*}Aut_K(F_1)&\longrightarrow Hom_K(F_1\cap F_2,K^{alg})\\ \rho&\longmapsto ...
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Finite number of maximal ideals above a maximal ideal in general integral extension [duplicate]

Let $A\subseteq B$ be an integral extension of integral domains and let $K$ and $L$ be the fields of fractions of $A$ and $B$ respectively. Assume that the field extension $L/K$ is Galois with finite ...
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Field extension with Galois group $C_2$

Given a field extension $E \subseteq F$ with the Galois automorphism group of $F:E$ having order 2, must $F:E$ necessarily be a normal extension? I remembered seeing this claim somewhere (I can't ...
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Is the Splitting Field necessarily a subset of a field where the polynomial splits?

Just a very basic theoretical question that has been puzzling me. Let $f(x)$ be a polynomial with coefficients in a field $F$. Let $K$ be the splitting field of $f$ over $F$. Say $f$ splits in a ...
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Is $\mathbb{Q}(\zeta_5)/\mathbb{Q}(\sqrt5)$ the maximal finite abelian extension of $\mathbb{Q}(\sqrt5)$ unramified away from $5\infty$?

The following problem appears in a homework question posed by B. Conrad (2(i) here: http://math.stanford.edu/~conrad/249BPage/homework/hmwk9.pdf): Using class field theory, prove that ...
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Galois Theory cyclic extension

Let $F\supset E$ be a Galois extension, and $a\not\in E$ an element of $F$ which is contained in every proper extension of $E$ contained in $F$. Prove that the Galois is nilpotent and cyclic. I can ...
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Showing that a third degree polynomial with roots in terms of roots of fourth degree polynomial has rational coefficients

Been sitting with this question from for a whole bunch of hours now. I'm studying for an exam so I don't want to be stuck for too long, however much I dig endlessly pondering. Let $f(t)$ be a ...
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f(x) is irreducible over K if and only if Gal(F/K) acts transitively on the roots of f(x).

Consider the following theorem from book John A.Beachy Abstract Algebra. Proposition 8.6.2. I don't understand why it can be extended to an automorphism of the splitting field $F$. Can someone ...
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Are the eigenvalues of a symmetric matrix continuous as function of the elements?

Let $A_\theta =((a^\theta_{i,j}))_{d\times d}$ be a real-symmetric matrix indexed by a vector $\theta \in \mathbb{R}^d$. Is $\theta \mapsto \lambda_{\max}(A_\theta)$, the maximum eigenvalue of ...
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Galois group for Kummer extension over Cyclotomic extension of $p$-adic field

I am trying to recover the Galois group of the extension $E/F$, where $E$ and $F$ are the fields defined below. $F$ is a finite extension of $\mathbb{Q}_p$, containing a primitive root of unity ...
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Irreducible of $\mathbb Z/p\mathbb Z$.

What are the irreducible number of $\mathbb Z/p\mathbb Z$ ? It looks strange since in a field it looks complicate to talk about irreducible since all element are invertible. So if my question has no ...
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Compute Galois groupe of $\mathbb F_q/\mathbb F_p$

I have to compute Galois group of $\mathbb F_q/\mathbb F_p$ where $q=p^n$. I know already that $\mathbb F_q/\mathbb F_p$ is galois, so I don't need to prove it. Moreover, I know that ...
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Limitations on the structures of normal subgroups and generating a n-degree polynomial formula

I was considering the problem of expressing the roots of a general polynomial $$ a_0 + a_1 x + ... a_n x^n$$ where $a_i, x \in \Bbb{C}$ Roots of course cannot be solely expressed using the field ...
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Why $f$ is invariant for $G_f$.

Let $F$ a field and $f\in F[X]$ a separable polynomial. Let $K_f$ the splitting field of $f$ and $G_f=Gal(K_f/F)$ its Galois group. We suppose it act transitively on the roots of $f$.$\alpha ...
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Galois group of X + XY + Y^n over Q(X)

I have seen that the Galois group of the polynomial $X + XY + Y^n$ over $Q(X)$ is $S_n$. I would appreciate a clear proof of this.