Questions tagged [galois-extensions]
For questions about Galois extensions of fields. We say that an algebraic extension $L/K$ is a Galois extension iff the subfield of $L$ that is fixed by automorphisms of $L$ which fix K is exactly $K$.
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Extending a field homomorphism on a finite field
Let $p$ be a prime number. Let $a,b,c$ be positive integers, $a$ divides $b$ and $b$ divides $c$.
Fix homomorphisms $f\colon\mathbb{F}_{p^{a}}\to\mathbb{F}_{p^{b}}$ and $g\colon\mathbb{F}_{p^{a}}\to\...
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I ask about the galois extension proof of the fundamental theorem
I was reading the book The Theory of Fields and Galois by J.S. Milne, and I have a question about a demonstration that is done in the book,
in THEOREM 3.16 (FUNDAMENTAL THEOREM OF GALOIS THEORY) in ...
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Is there an example, such that $\omega\in F(\beta)\land \omega\notin F$?
$F$ is perfect, $E=F(\beta)$, where $\beta^n=a\in F$, then $E$ is a Galois extension over $F$ if and only if the field contains the n-th primitive root $\omega$, where $\omega^n=1$
Question: I am ...
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$\Bbb Q \subseteq \Bbb Q(\sqrt {a + \sqrt b})$ is normal iff at least one of $a^2 − b$ and $(a^2 − b)/b$ is a square in $ \Bbb Q$
Let $a, b$ be integers such that $b$ is not a square. Show that the field extension $\Bbb Q \subseteq \Bbb Q(\sqrt {a + \sqrt b})$ is normal if and only if at least one of $a^2 − b$ and $(a^2 − b)/b$ ...
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$b$ not a square, $a^2 − b$ a square Why is $\Bbb Q(\sqrt {a + \sqrt b})$ the splitting field of $f(x)=(t^2-a)^2-b$ and a normal extension?
Let $a; b$ be integers such that $b$ is not a square. if $a^2 − b$ is a square in $ \Bbb Q$.,
then the field extension $\Bbb Q \subseteq \Bbb Q(\sqrt {a + \sqrt b})$ is normal.
My try: If $a^2 − b$ ...
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Ramification of primes on $\overline{\mathbb Q}$
My concern arises about the existence of unramified primes in infinite extensions of $\mathbb Q$, particularly in $\overline{\mathbb Q}$.
Now, in general there exist be a definition of ramification ...
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How Do I Calculate The Galois Group And Intermediate Fields Of A Certain Extension Using Magma?
here MAGMA Commands for Galois Theory calculations it is discussed how to calculate the galois group when the the field that is fixed is say the rationals. But what if the field that is fixed is $\Bbb{...
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Splitting field of a separable (and irreducible) polynomial is separable
I was struggling proving this and didn't manage to find any solution here that felt understandable for my level, so I am submitting my best idea, which seems right.
Let $L$ be a splitting field of a ...
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How to generate all elements of an extension field from a base field? (GF(2))
I am trying to understand how to show something is a primitive polynomial; I understand it has to be irreducible by definition, and according to Wolfram:
...
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Action of Galois group on scheme
The following exercise is the 2.10 page 97 of Liu's book. Let $k/K$ a finite Galois extension and $X$ a $k-$scheme of finite type over $k$ (i.e. a $k$ algebraic variety).
I have the following ...
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Elements with norm $1$ in a quadratic field [closed]
Consider the number field $K=\mathbb{Q}(\sqrt{5})$ and the norm map $N:\mathbb{Q}(\sqrt{5}) \to \mathbb{Q}^*$ given by $N(a+b \sqrt{5})=a^2-5b^2$.
Is there an explicit description of the kernel of ...
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Determine the degree of a root of $X^p-X-\alpha$ (Serge Lang Algebra exercise VI.29)
Let $K$ be a cyclic extension of a field $F$, with Galois group $G=\langle \sigma \rangle$ and assume that $\operatorname{char}F=p$ and that $[K:F]=p^{m-1}$ for some $m>1$. Let $\beta$ be an ...
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Prove a Weaker Version of the Normal Basis Theorem
Question. For a finite Galois cyclic extension $E/F$, prove that there is a primitive element $\alpha$ such that $$\sum_{\sigma\in\operatorname{Gal}(E/F)}u_{\sigma}\sigma(\alpha) = 0$$ for $u_{\sigma} ...
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Modular equations admitting a solution
In a finite Field $\mathbb{F}_q$ with $q = p^n$ and $p$ a prime numbers I am interested in the modular equation
$$
x^2+3 \equiv 0 \pmod p.
$$
More precisely I would like to ask whether there is a ...
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$L^{⟨Gal(L/F),Gal(L,F’)⟩}=F \cap F'$
Let $L/k $ be a Galois extension. If $F$ and $F'$ are intermediate fields of $L/k$ then
$L^{⟨Gal(L/F),Gal(L,F’)⟩}=F \cap F'$
($⟨G;G′⟩$ is used to denote the subgroup generated by $H ∪ H′.$)
Using some ...
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Intermediate fields of a Galois extension
Let $K$ be the splitting field of $f(X)=X^3-2$ over $\mathbb{Q}$. I am asked to find complete list of intermediate fields $k$, $\mathbb{Q}\subseteq k\subseteq K$ such that $[k:\mathbb{Q}]=3$.
I've ...
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Describe the Galois group of a cubic and quartic over $\mathbb{F}_p(y)$
I'm studying "Note on a problem of Chowla" by Birch and Swinnerton-Dyer, it's about counting the number of values attained by a polynomial in $\mathbb{F}_p$.
In the paper, for a polinomial $...
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Generators of a Galois group in SAGEMath
I have a (potentially silly) question about a problem I've been having with SAGEMath.
Let $K$ be the maximal totally real subfield of the cyclotomic field of conductor $24$, i.e. the number field with ...
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Pove that the Galois group of $p(x) = f(x^2)$ is not abelian
Let $f(x) \in \mathbb Q[x]$ be an irreducible cubic polynomial with three real roots $\alpha, \beta, \gamma \in \mathbb R$ such that $\alpha$ is
negative and $\beta$ is positive. Prove that the Galois ...
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Example of a field on which every irreducible polynomial has degree a power of $p$
Exercise A-47 in Milne's Fields and Galois Theory notes asks to prove that if $p$ is a prime number and $F$ is a field of characteristic zero such that every irreducible polynomial $f(X)\in F[X]$ has ...
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Compute $[ \mathbb Q(\zeta_8) : \mathbb Q(i)]$ and find a basis for the extension.
This is a follow-up to my previous question; I wanted to give a different example to make sure I understand what is going on.
Compute $[ \mathbb Q(\zeta_8) : \mathbb Q(i)]$ and find a basis for the ...
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Compute degree of $\mathbb Q(\zeta_8)$ over $\mathbb Q(\sqrt 2)$ and find a basis.
This is a review problem for a Galois theory course.
Compute $[\mathbb Q(\zeta_8): \mathbb Q(\sqrt 2)]$ and find a basis. (Where $\zeta_8$ is the primitive $8$th root of unity).
I compute the degree ...
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Galois group of $x^4 +3 \in \mathbb Q[x]$ using Kaplansky's Theorem
This is exercise 4.5 from Baker Galois Theory:
Use Kaplansky's theorem to find the Galois group of the splitting field $E$ of the polynomial $x^4 +3 \in \mathbb Q[x]$ over $\mathbb Q$. Determine all ...
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Product vs. join of subgroups of a finite group
Hypotheses:
$G$ is a finite group
$A$ and $B$ are subgroups if $G$.
Definitions:
$AB = \{ ab \mathbin{|} a \in A, b \in B \}$ is the product of $A$ and $B$
$A \vee B$ is the join of (subgroup of $G$...
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if you have an extension that is not Galois, can you compose it with another extension and make it Galois? [closed]
I have $\mathbb Q (\sqrt[3]{5},\sqrt{2})$. I know that $\mathbb Q(\sqrt[3]{5})$ is not a Galois extension because although the splitting field of the polynomial $x^3 -5$ over $\mathbb Q$ is $\mathbb Q(...
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Why is the set of all embeddings of $K$ to $L$ a set of coset representatives for $H$ in $\text{Gal}(L/F)$?
In Exercise 14.2.17, Dummit & Foote:
Let $K/F$ be any finite separable extension, and let $\alpha\in K$. Let $L$ be a Galois extension of $F$ containing $K$ and let $H\leq \text{Gal}(L/F)$ be the ...
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When is a norm of a formal power series over a local field a polynomial?
Let $K$ be a finite extension of $\mathbb Q_p$ and $L/K$ be a finite Galois extension. Then also $L(T)/K(T)$ and $L((T))/K((T))$ are Galois extensions with Galois group isomorphic to ${\rm Gal}(L/K)$ ...
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Issues understanding $|\text{Gal}(E/F)| = [E:F] \iff E/F$ is Galois.
$\newcommand{\Gal}{\text{Gal}}$
Precursor Definitions: A finite field extension $E/F$ is called Galois if it is both normal and separable. The Galois group of $E/F$ is
$$
\Gal(E/F):= \{\sigma: E\...
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What is an Archimedean prime?
I need to learn some infinite ramification theory and I am stuck with understanding it. I understand that we consider the order of the inertia group as the ramification index, and if the inertia group ...
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Show a field extension is normal, then find the Galois group
Let $K=\mathbb{Q}(2^{\frac{1}{4}}, i)$. Show that $K$ is normal over $\mathbb{Q}(i)$, and $Gal(K/ \mathbb{Q}(i))$ is cyclic of order 4.
What I have done:
I showed that $K$ is the splitting field for $...
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Find a Galois extension whose Galois group is $\mathbb{Z}_4 \times \mathbb{Z}_4$
In my notation, $\mathbb{Z}_n$ = the integers modulo $n$.
Find a Galois extension whose Galois group is $\mathbb{Z}_4 \times \mathbb{Z}_4$. (The additive group).
I thought about using cyclotomic ...
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Suppose $K$ is a field such that every finite Galois extension of $K$ is cyclic. Is there a name for this property?
Let $K$ be a field such that the following property is satisfied: every finite Galois extension of $K$ is cyclic. Is there a name for this kind of fields or this property? I haven’t been able to find ...
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Question on the discriminant of a irreducible polynomial
Let $\mathbb{K}_5$ be the field with five elements. I don't know to describe the splitting field of $f(x) = x^{3} + x + 1\in\mathbb{K}[5]$ in terms of a root and the discriminant. Clearly, $f(x)$ is ...
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Quaternion division rings in characteristic $0$
Hamilton's original quaternions $\mathcal{Q}$ form a division ring which is $2$-dimensional over $\mathbb{C}$, and which has $\mathbb{R}$ as its center. Define its elements as $a + bi + cj + dk$, ...
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Algebraic numbers with special properties
There is a "result" which would help me in my research.
Lemma. Let $d$ and $n$ be positive integers and let $\alpha$ be a real algebraic number with degree $r$. Then, there exists a real ...
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Why isn't the splitting field always Galois?
My question stems from the proof of the fundamental theorem of algebra in Dummit and Foote. There they claim that for any real polynomial $f(x)$ with roots $\alpha_1, \dots, \alpha_n$, $\Bbb R(\...
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Proof about the closed integral $\mathbb{Z}_p[\zeta_p]$
Assuming $\zeta_p$ is a root of unit, I need to show that $\cal{O}$$(\mathbb{Q}_p(\zeta_p))=\mathbb{Z}_p [\zeta_p]$. Where $\cal{O}$$(F)=${$x\in F:|x|_p\le 1$} and $\mathbb{Z}_p$ is the ring of p-adic ...
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The relation between $Gal(Z_2(M)/Z_1(M))$ and $Gal(Z_2/Z_1)$
Let $L/K$ be a field extension with $K \subset Z_1,Z_2 \subset L$. And furthermore let $M$ be a set completely disjoint with $K,Z_1,Z_2, L$.
If we have $\phi \in Gal(Z_2(M)/Z_1(M))$, then $\phi|_{Z_2} ...
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Conditions on $K(X)/K(X^n)$ being Galois
Statement: $K(X)/K(X^n)$ is Galois if and only if $p$ doesn't divide $n$ and $K$ contains all the zeroes of the separable polynom $g := X^n-1 \in K[X]$. (the polynom is certainly separable because $p$ ...
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Any normal third degree extension of $\mathbb Q$ contained in $\mathbb C$ must be contained in $\mathbb R$
We let $K\subset \mathbb C$ be a finite normal field extension of $\mathbb Q$ such that $[K:\mathbb Q]=3$. Show that $K\subset \mathbb R$.
I know that no cyclotomic extension of $\mathbb Q$ will do ...
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Extensions of field embeddings.
Let $L/K$ be a normal extension of number fields and let $\sigma: $K$ \rightarrow \mathbb{C}$ be an embedding of $K$. We know that there are $[L:K]=n$ extensions of $\sigma$ to L.
$\sigma$ is a real ...
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Cyclic Galois group and field norm [duplicate]
I am struggling on solving the problem related to Galois theory.
Let $K/F$ be a cyclic Galois extension whose Galois group is generated by $\sigma$ of order $n$. Suppose that $\alpha \in K$ satisfies ...
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Galois group of the quintic $x^5 -11x + 1$.
Consider $f = x^5 -11x + 1 \in \mathbb{Q}[x]$. I want to prove that its not solvable by radicals. I know that its solvable by radicals iff its galois group is solvable. My attempt was first to use the ...
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Show there is a tower of cyclic field extensions of prime degree from $\mathbb{Q}(\sqrt[12]{5})$ to $\mathbb{Q}$
Of course, it suffices to find tower of Galois extensions of prime degree, as these would have to be cyclic. My first thought was to try extending $\mathbb{Q}$ first by $\sqrt 5$, then $\mathbb{Q}(\...
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I am proposing two conjectures regarding the BCH codes ! Can anybody prove or disprove my conjectures?
I was working with BCH codes for Compressed Sensing. Through the simulation of BCH codes, I found some interesting facts which are useful for my Ph.D. thesis. I don't have mathematical proof of my ...
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Galois group of $x^n+x+1$
I’m struggling with a following question. Having a polynomial $P(x) = x^{10} + x + 1 \in \mathbb{Q}[x]$ find its Galois group. The question for $x^{3} + x + 1$ is much easier for me, because, all the ...
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Proving there are no intermediate normal extensions
Let $E/F$ be a finite extension and $L \supseteq E$ an splitting field of a polynomial $g \in F[x]$ such that every irreducible factor of $g$ in $F[X]$ has a root in $E$. I need to prove $L/F$ is a ...
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Help understanding why an automorphism is uniquely determined
Im reading a proof of the fundamental theorem of Galois theory and I'm having trouble understanding the proof of the following part:
Let $F/K$ be a finite Galois extension with $G = \operatorname{Gal}...
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Effective degree bound for solvability by radicals
Let $P\in{\mathbb Q}[X]$ be an irreducible polynomial of degree $n\geq 3$, and let $\mathbb L$ be the decomposition field of $P$. Denote the Galois group of the extension ${\mathbb L}:{\mathbb Q}$ by $...
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Why isn't the crossed product algebra $(K(\sqrt{a}), Gal(K(\sqrt{a})/K, k)$ independent of a?
I am trying to work through an example to understand the correspondence between $H_2(Gal(K(\sqrt{a})/K),K(\sqrt{a}))$ and $Br(K(\sqrt{a}),K)$. Here is my confusion:
Let $E = K(\sqrt{a})$ be a ...
