All questions with tag [math: galois-theory]


Field Extension problem beyond $\mathbb C$

There are great deals of areas in between $\mathbb C$ and also Meromorphic Functions on $\mathbb C$. As an example set of "All Even Meromorphic Functions on $\mathbb C$" is a subfield in between $\mathbb C$ and also Meromorphic Functions on $\mathbb C$. Inquiry: How to classify such subfields? I have no suggestion whether someone res...
2022-07-25 20:47:10

Abel-Ruffini Theorem Clarification

Let $n \geq 5$. I want to show that $\exists p\in \mathbb{Q}[X]$ of degree $n$ with roots which are not possible to find via radicals and rational functions from the coefficients of $p$. I've got the following theorem (Abel-Ruffini) written down in my notes. (1) $\exists$ transcendental $\alpha_1,\dots\alpha_n\in\mathbb{C}$ s.t. $F=\mathbb{Q}(\a...
2022-07-25 20:44:21

Solubility of a Galois Group

going over some past papers with no answers and would like a bit of help if possible.. I've shown that for p a prime number then $x^p-1 \in K[x]$ is abelian where K is a subfield of $\mathbb{C}$. I've now been asked to show that the Galois group of $x^p-a$ over K is soluble with $a \neq 0 \in K$. I know any abelian group A is soluble, since ${...
2022-07-25 20:43:41

Find all finite fields k whose subfields form a chain: if $k'$ and $k''$ are subfields of $k$, then either $k' \subseteq k''$ or $k'' \subseteq k'$.

Find all finite fields $k$ whose subfields form a chain: that is, if $k'$ and $k''$ are subfields of $k$, then either $k' \subseteq k''$ or $k'' \subseteq k'$. So, I understand that I'm trying to find the values of $n$ such that the subfields of $\mathbb{F}_{p^n}$ (where $\mathbb{F_{p^n}}$ is the Galois field of order $p^n$) form a chain. Howe...
2022-07-25 20:20:51

Two Equivalent Notions of Algebraic Simple Extension (Proof)

When reviewing messages concerning area expansions I've found the adhering to 2 interpretations for the straightforward expansion $K(\alpha)$ where $\alpha$ some algebraic number over $K$. They are (1) $K(\alpha)$ is the tiniest area having both $K$ and also $\alpha$ (2) $K(\alpha)=\{\sum_{0}^{\infty}k_i\alpha^i:k_i\in K\}$ I have not had t...
2022-07-25 19:43:36

Prove that $\mathrm{Aut}(L(\zeta_n)/K(\zeta_n))$ is a subgroup of $\mathrm{Aut}(L/K)$

Let $L/K$ be a Galois extension of fields and let $\zeta_n$ be a primitive $n$th root of unity in some field extension of $L$, where $n$ is not divisible by the characteristic. Prove that $\mathrm{Aut}(L(\zeta_n)/K(\zeta_n))$ is a subgroup of $\mathrm{Aut}(L/K)$. I'm unsure because I don't really know what is required. If $\sigma \in \mathrm{Aut...
2022-07-25 19:40:55

Proof that a polynomial is irreducible in $\mathbb{Q}$

Let $p$ be a prime number, and let $m,k_1,\ldots,k_{p-2}$ be even numbers. Define the polynomial $h(x)=(x^2+m)(x-k_1)\cdots(x-k_{p-2})$ and $r=\min \{|h(a)|\mid a\in\mathbb{R},h'(a)=0\}$. Under these condition I've proven that $r>0$. Now let $n$ be an odd number large enough so that $\frac{2}{n}<r$ and let $f(x)=h(x)-\frac{2}{n}$, ...
2022-07-25 19:39:35

Why is this extension of Galois?

Let $F$ be a subextension of $\mathbb{C}$ maximal with respect to not containing $\sqrt2$. Let $K/F$ be a finite extension with $K\subset\mathbb{C}$. Then $K/F$ is of Galois and $[K:F]$ is a power of a prime. Could you help me solve this exercise? Following the idea of Olivier: Let $L$ be the normal closure of $K/F$. If $K\neq F$, then $\sqrt2\i...
2022-07-25 16:30:47

Calculating fixed field of a particular action

Let $K = \mathbb F_p(x)$, and let $H = \left\{\begin{pmatrix} d & a \\ 0 & 1 \end{pmatrix} \ \big| \ a \in \mathbb F_p, d \in \mathbb F_p^\times\right\}$ be a group under multiplication which acts on $K$ via $\begin{pmatrix} d & a \\ 0 & 1 \end{pmatrix} x = dx + a$. How can I find the fixed field of $H$? If $f =...
2022-07-25 16:28:44

Is the size of the Galois group always $n$ factorial?

I am researching field theory, and also I simply began the phase on Galois theory. Given that a Galois expansion is the splitting area of some polynomial $p(x)$ and also this polynomial have specifically $n$ origins in the expansion area I assume that every automorphism of the expansion permutes the origins of $p(x)$ [and for every single permu...
2022-07-25 16:15:01

Why $H\neq N_G(H)$?

Let $K$ be a field, $f(x)$ a separable irreducible polynomial in $K[x]$. Let $E$ be the splitting field of $f(x)$ over $K$. Let $\alpha,\beta$ be distinct roots of $f(x)$. Suppose $K(\alpha)=K(\beta)$. Call $G=\mathrm{Gal}(E/K)$ and $H=\mathrm{Gal}(E/K(\alpha))$. How can I prove that $H\neq N_G(H)$? My idea was to take a $\sigma: E\rightarrow\ba...
2022-07-25 15:52:23

Example of field $K$ with $\mathrm{char}(K) > 0 $, such that $[K(\alpha):K] = [K(\beta):K]$ but $K(\alpha) \not \cong K(\beta)$

I'd like to fine an example of field $K$ and elements $\alpha, \beta$ such that $\mathrm{char}(K) = p> 0 $, $[K(\alpha):K] = [K(\beta):K]$ but $K(\alpha) \not \cong K(\beta)$. This obviously can't work if $K$ is a finite field. So I need to find a non-finite $K$. The only ones that pop into my head are $\mathbb F_p(t)$, $\mathbb F_p(t^p)$...
2022-07-25 15:45:29

Show the norm map is surjective

Let $K/F$ be an extension of finite field. Show that the norm map $N_{K/F}$ is surjective. Here is what I have so far: Since $F$ is a finite field and $K/F$ is a finite extension of degree $n$, so $\operatorname{Gal}(K/F)=\langle\sigma\rangle$, where $\sigma(a)= a^{q}$ with $q=p^{m}=|F|$. In addition, by primitive element theorem, $K=F(\alpha)$ ...
2022-07-25 15:39:59

Galois Group of $(x^3-5)(x^2-3)$

I am having some trouble calculating the Galois group (over $\mathbb{Q}$) of $(x^3-5)(x^2-3)$. I can see the splitting field is $F:=\mathbb{Q}(\sqrt[3]{5},\omega,\sqrt{3})=\mathbb{Q}(\sqrt[3]{5},i,\sqrt{3})$, where $\omega$ is a primitive 3rd root of unity, and it has degree 12 over $\mathbb{Q}$. Since the extension is Galois (it is a splitting ...
2022-07-25 15:37:37

What is a maximal abelian extension of a number field and what does its Galois group look like?

How does one know that a number field $K$ has a maximal abelian extension (unique up to isomorphism) $K^{\text{ab}}$? I've read proofs involving Zorn's lemma that it has an algebraic closure (And that algebraic closures are unique up to isomorphism.) $\bar{K}$ All these proofs involved ideals of the polynomial ring in variables $x_f$, $f$ an i...
2022-07-25 15:34:03

Need help determining the Galois group of an extension

In an assignment I got I have been asked to try to determine when $E/F$ is a Galois Extension, and determine the Galois group of such an extension. $\textbf{Context:}$ $F$ is any field of characteristic zero and $E = F(\sqrt{c},\sqrt{a + b\sqrt{c}})$, where $a,b,c \in F$, $c$ is not a square in $F$ and $a + b\sqrt{c}$ is not a square in $F(\sqr...
2022-07-25 10:54:03

Endomorphisms and Automorphisms

Let $L$ be a field extension of $K$. Consider the set $\operatorname{End}_KL$ of all functions from $L$ to $L$ which are linear over $K$. A subset of $\operatorname{End}_KL$ is $\Gamma(L:K)$, the group (under composition) of all automorphisms on $L$ which fix $K$. So, we can consider $\operatorname{End}_KL$ to be a vector space over $K$. We h...
2022-07-25 10:50:14

Order and generator of the Galois group of an extension of finite fields

I'm searching for the order and also define a generator of the team $$\mathrm{Aut}_{\mathrm{GF}(2^3)}(\mathrm{GF}(2^{12}))$$ It is clear that the order is 4, yet just how would certainly you define the generator? Many thanks!
2022-07-25 10:49:09

An example that the Galois correspondence fails if the extensions is not Galois

Let $F/K$ be a finite extension of fields. $S$ the set of subgroups of $\mathrm{Aut}_K(F)$ and $I$ the set of intermediate fields of the extension $F/K$. Define the function $\varphi:S\rightarrow I$ as $\varphi(G)= F^G$ where $F_G$ is the subfield of $F$ fixed by $G$. Could you help me to find an example of extension $F/K$ where $\varphi$ is not...
2022-07-24 09:41:19

When the group of automorphisms of an extension of fields acts transitively

Let $F$ be a field, $f(x)$ a non-constant polynomial, $E$ the splitting field of $f$ over $F$, $G=\mathrm{Aut}_F\;E$. How can I prove that $G$ acts transitively on the roots of $f$ if and only if $f$ is irreducible? (if we suppose that $f$ doesn't have linear factor and has degree at least 2, then we can take 2 different roots that are not in $F...
2022-07-24 09:40:58