Use this tag for questions about fields and field theory in abstract algebra. A field is, roughly speaking, an algebraic structure in which addition, subtraction, multiplication, and division of elements are well-defined. Please use (galois-theory) instead for questions specifically about that ...

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How do we conclude that $K(a,b) \subseteq \mathcal{A}_{E|K}$?

Let $K \leq E$, $\mathcal{A}_{E|K}=\{a \in E \text{ with } a \text{ algebraic } |K\}$ $K \subseteq \mathcal{A}_{E|K} \subseteq E$ We claim that $\mathcal{A}_{E|K}$ is a field. $a, b \in ...
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3answers
25 views

Solutions of $x^{p^n}=x$ form a subfield $F\subset K$.

$K$ is an algebraically closed field such that Char $K=p$. Show that the solutions of $x^{p^n}=x$ form a subfield $F\subset K$. My Work: $0$ is a solution and $0\in K$. Let $\alpha$ be a non zero ...
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3answers
121 views

How many solutions of the equation $x^2-y^2=1$ are there with $x,y\in\mathbb F_{p^n}$?

Let $p>2$ be an odd prime. Let $\mathbb F_{p^n}$ be the field with $p^n$ elements. How many solutions of the equation $x^2-y^2=1$ are there with $x,y\in\mathbb F_{p^n}$? My work: Char $F=p$. ...
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2answers
41 views

If $K \leq L$ a finite extension then it is algebraic.

I am looking at the proof of If $K \leq L$ a finite extension then it is algebraic. The proof is the following: Let $[L:K]=n<\infty$. Let $a \in L$. We will show that $\exists$ a ...
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1answer
21 views

Find the minimal irreducible polynomial $Irr(a, \mathbb{Q})$

Let the field extension $\mathbb{Q} \leq \mathbb{C}$ and $a=e^{\frac{2 \pi i}{8}}$. I have to find the minimal irreducible polynomial $Irr(a, \mathbb{Q})$. $a$ is a root of the polynomial ...
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Finding a Galois group using a cubic resolvent

I've heard there's a fast way to find the Galois group of a quartic polynomial using its resolvent. Can anyone explain how it's done or give a reference? Is this method the same as the general ...
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2answers
45 views

How to prove the polynomial $ x^{2011}-x\in \mathbb{Z}_{2011}[x] $ is separable?

I am trying to prove that the polynomial $ x^{2011}-x\in \mathbb{Z}_{2011}[x] $ is separable. Definition: Let K be a field and $f\in K[x]$ an irreducible polynomial. The polynomial $f$ is said to be ...
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35 views

Integral domain - Embedding

Let $R$ be an integral domain and the homomorphism \begin{align} \phi\colon \mathbb{Z} &\rightarrow R \\ n &\mapsto n \cdot 1_R \end{align} What does it mean that if $\ker \phi =\{0\}$ then ...
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2answers
34 views

What are the intermediate fields of $\mathbb Q(\sqrt[3]2,e^{\frac{2i\pi}{3}})$ (Galois group)

The elements of Galois group are \begin{align*} \sigma _1:\mathbb Q[\sqrt[3]2,e^{\frac{2i\pi}{3}}]&\longrightarrow \mathbb Q[\sqrt[3]2,e^{\frac{2i\pi}{3}}],\\ \sqrt[3]{2}&\longmapsto ...
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1answer
50 views

Existence of a group isomorphism between $(\mathbb K,+)$ and $(\mathbb K^\times,\cdot)$

Let $(\mathbb K,+,\cdot)$ be a field. Is there a group isomorphism between $(\mathbb K,+)$ and $(\mathbb K^\times,\cdot) $ ? The answer should clearly be negative. I tried to proceed via ...
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2answers
66 views

Adjoining a number to a field

When I studied algebra, we talked about fields such as $\mathbb{Q}[\sqrt{2}]$, the rational numbers with the square root of two adjoined to the field. Structures like these are called field extensions ...
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32 views

Transcendence degree of polynomials equals the transcendence degree of the function field of the polynomials

How to prove that transcendence degree of the function field $k(f,g)$ is equal to the transcendence degree of $\{f,g\}$ where $f$ and $g$ are two multivariate polynomials? If $\{f,g\}$ are ...
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1answer
74 views

Subfield of $\mathbb{R}$ such that $\Bbb R/K$ is finite.

Is there a field $K \subset \mathbb{R}$ such that $1 < [\mathbb{R} : K] < \infty$? i.e a proper subfield of $\mathbb{R}$ such that the field extension $\mathbb{R}/K$ is finite.
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40 views

The number of automorphisms of a finite field

Let $M$ be a finite field and $|M| = p^s$, where $p$ is prime and $s \in \mathbb N$. Prove that the number of different isomorphisms field $M$ to $M$ equal to $s$ and this isomorphisms form a cyclic ...
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1answer
40 views

The finite field extension

Let field $K$ embedded into the finite field $M$. Prove that $M = K(\theta)$ for some $\theta \in M$. I have tried 2 ways but got stuck at both. 1) Let $|K| = p^s$ and $|M| = p^{st}$ for prime $p$ ...
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1answer
26 views

Algebraically independent equivalent conditions

I have some problems to understand the field extensions. Namely, Let $K$ be a field and $E$ its extension. Let $x_1,\ldots ,x_n$ in $E$ and $0<k<n$. Show that TFAE Family $(x_1,...,x_n)$ is ...
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3answers
79 views

Is it obvious that $\mathbb Q(\sqrt 3+\sqrt 5)=\mathbb Q(\sqrt 3,\sqrt 5)$? [duplicate]

Is it obvious that $\mathbb Q(\sqrt 3+\sqrt 5)=\mathbb Q(\sqrt 3,\sqrt 5)$ ? If not how can I show it ?
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1answer
34 views

Transcendence degree of the ring generated by two algebraically dependent polynomials [on hold]

How to prove that If two polynomials $f$ and $g$ are algebraically dependent, then any two polynomials in the ring generated by $f$ and $g$ would be algebraically dependent?
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1answer
46 views

Degree of field extension question [closed]

find the degree of the field of extension $\Bbb Q(\sqrt{2},\sqrt[4]{2},\sqrt[8]{2})$ over $\Bbb Q$. 1) 4 2) 8 3) 14 4) 32 I think it is 8.
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2answers
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Finding Galois group of $x^6 - 3x^3 + 2$

I'm trying to find the Galois group of $$f(x)= x^6 - 3x^3 + 2$$ over $\mathbb{Q}$. Now I can factorise this as $$f(x) = (x-1)(x^2 + x + 1)(x^3 - 2)$$ I can see the splitting field must be ...
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1answer
22 views

Maximal algebraically independent subset and transcendence basis

I'm studying transcendence basis and I got stuck with the following problem: Let $K$ be a field and $E$ its extension. Let $S$ be a subset of $E$ such that $E$ is algebraic with respect to $K(S)$. ...
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0answers
55 views

When is $F(x+y) = F(x,y)$ for field $F$?

If $F$ is a field and $x,y$ are in an algebraic extension of $F$, I'm curious as to what we can say about $[F(x+y):F]$. I can easily prove the following:   $[F(x+y):F] \mid [F(x,y):F]$   ...
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2answers
58 views

Does there exist a subfield $S$ of $\mathbb C$ such that $\mathbb R \subset S \subset \mathbb C$?

Does there exist a subfield $S$ of $\mathbb C$ such that $\mathbb R \subset S \subset \mathbb C$ ? ; I kind of have a feeling that there does not exist any such $S$ but cannot prove . Thanks in ...
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0answers
37 views

Irreducibility of a polynomial with algebraically independent coefficients

I am learning some kind of field theory. Let $\mathbb{Q}'$ be the smallest subfield in $\mathbb{C}$ containing all roots of unity. Recently I read a book on Galois theory and met the following ...
2
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0answers
32 views

Uniqueness of an embedding theorem for Real differential fields

I will follow a preliminary exposition for the problem in question, which will essentially follow the format on http://www4.ncsu.edu/~singer/papers/model_diff_fields.pdf [pg. 87]: Let $K$ be a real ...
2
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0answers
84 views

A question about field extension: Zariski's lemma

Suppose $E$ is a field extension of $F$ and there exists $\alpha_1,\alpha_2,\ldots,\alpha_n\in E$ such that $E=F[\alpha_1,\alpha_2,\ldots,\alpha_n]$, then the field extension $E/F$ is algebraic. Is ...
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1answer
23 views

Prove algebraic closure

Let $L/K$ be a field extension such that $L$ is algebraically closed. Show that $\{a\in L\mid[K(a):K]\lt\infty\}$ defines an algebraic closure of $K$. So this is the set of minimal polynomials ...
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2answers
30 views

Linear composition

can you help me with this quest? About composition $f$ and vector space $\mathbf{V}=\mathbb{Z^4_2}$ we know the following: $f \circ f = id_V$,$~~f $ $ \left(\begin{array}{ccc} 1\\ 0\\ 1\\ 0\\ ...
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1answer
71 views

A question on the relation of the Galois group as field automorphisms and the Galois group as permutations of roots

Let $f\in K[X]$ be a monic separable polynomial and $L$ a splitting field of $f$. Let $M=\{l_1,\ldots,l_n\}$ be the set of roots of $f$ in $L$, i.e. $$ f=(X-l_1)\cdots(X-l_n). $$ The Galois group ...
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1answer
16 views

Finiteness of a simple extension

Here I have two propositions from p.521 on Abstract Algebra written by Dummit Foote. Let $\alpha$ be algebraic over the field $F$ and let $F(\alpha)$ be the field generated by $\alpha$ over $F$. ...
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If there exists a cyclic extension of degree $p$ of $K$, then there exists a cyclic extension of degree $p^n$.

If $char K=p \neq0 $, let $K_{p}=\{ u^p-u : u\in K\}$. Show that if there exists a cyclic extension of degree $p$ of $K$, then there exists a cyclic extension of degree $p^n$ for every $n \geq 1$. ...
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1answer
21 views

Does the fixed field of automorphisms group characterize Galois extensions?

If $E/K$ is a field extension we use the notation $\def\Aut{\operatorname{Aut}}\Aut(E/K)$ for the set of field automorphisms of $E$ that are the identity over $K$. It's immediate that the set ...
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0answers
51 views

If $F$ is finite over $L_1$ and $L_2$, is it also finite over $L_1 \cap L_2$?

Let $F$ be a field and $L_1$, $L_2$ two subfields such that $F$ is finite over both $L_1$ and $L_2$. Is $F$ necessarily finite over the intersection $L_1 \cap L_2$?
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1answer
44 views

Show that $E=\mathbb{Q}(a)$

Let $f(x) \in \mathbb{Q}[x]$ an irreducible polynomial with the splitting field $E$ and let the group $Gal(E/\mathbb{Q})$ be abelian. If $a$ is a root of $f(x)$ then $E=\mathbb{Q}(a)$. Could you ...
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31 views

constructibility of a number

Following task: If $\alpha$ and $\beta$ are algebraic numbers, having the same minimal polynomial, then I should show that $\alpha$ is constructible if and only if $\beta$ is constructible. I don't ...
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1answer
46 views

Is $\mathbb{Q}(\pi)$ a simple extension of $\mathbb{Q}\left(\frac{\pi^3}{1+\pi}\right)$?

In the case of an algebraic extension, I could think easier than this case. But I got stuck in this problem. I know that the dimension of $\mathbb{Q}(\pi)$ over ...
2
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3answers
47 views

How can I find the degree of the extension?

Let $\omega_7=e^{2\pi i/7}$ . How can I find the degree of the extension $\mathbb{Q} \leq \mathbb{Q}(\omega_7+\omega_7^5)$?? Could you give me some hints??
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1answer
19 views

About finite field extensions and their generators

I am with trouble to prove this statment: "Suppose that $\{\beta_1,...,\beta_n\}$ is a base of $L|K$ and $\mathcal{M}$ is a subset of some fied $M\supseteq L$. Prove that $\{\beta_1,...,\beta_n\}$ ...
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1answer
31 views

Cyclotomic extension of $\mathbb{F}_p((T))$

I feel very confused about why adding n-th roots of unity to $\mathbb{F}_p((T))$ would give $\mathbb{F}_{p^n}((T))$. (Is this true?)
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1answer
26 views

Lattice basis for prime divisor of $(p)$ [closed]

Suppose that $d \equiv 2$ or $3$ modulo $4$, and that a prime $p \neq 2$ does not remain prime in $R$. Let $a$ be an integers such that $a^2 \equiv d$ modulo $p$. How would I go about showing that ...
2
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1answer
19 views

General construction of $\operatorname{GF}(2^k)$

Is there a general method of constructing fields of the form $\operatorname{GF}(2^k)$? (preferably something that is easily manipulable by a computer.) I know that one can look for an irreducible ...
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2answers
57 views

Infinite algebraic extension of a finite field

I have recently started studying algebraic field extensions and I got to know that algebraic closures $\overline{F}$ of finite fields $F$ are infinite. Therefore, I've asked myself the following ...
2
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1answer
35 views

The normal closure of a field extension

I'm making my first steps in abstract algebra and I was wondering, if there is a technique to determine the normal closure of a given extension, cause all I know is a theoretical definition: $K_n$ is ...
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0answers
30 views

Determine which roots of unity have degree at most 3

I need help to do exercises on "Abstact Algebra": 1.Determine all integer $n$, such that $\phi_n$ has degree at most $3$ over $\mathbb Q$, where $\phi_n=e^\frac{2\pi i}{n}$ .
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1answer
14 views

Approximating a field by perfect fields.

Let's consider an arbitrary field $K$ and raise the following question: in which sense can we approximate $K$ by a perfect field? Any reasonable notion of approximation by a perfect field should admit ...
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34 views

A question concerning cyclic field extensions.

In the study of cyclic extensions we have the following theorem: Theorem Let $K$ be a field containing an $n$-th primitive root of unity $\zeta$. Then the following claims hold: If ...
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10 views

If k>0 is a positive integer and p is any prime, show that Zp[√k]={a+b√k | a,b∈Zp} is a field if there doesn't exist x in Zp, such that x^2=k.

However, if there exist x in Zp such that X^2=k, then √k is in Zp and hence Zp[√k] is just Zp which is a field. What's my error? I am confused.
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1answer
25 views

Question regarding algebraicity of two elements whose sum and product are algebraic.

Let $\alpha , \beta \in \Bbb C$ and suppose $\alpha + \beta$ and $\alpha \beta$ are algebraic over $\Bbb Q$. Prove $\alpha , \beta$ are algebraic over $\Bbb Q$.
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2answers
25 views

The number of one-dimensional vector spaces in a field

Let $p$ be a prime number, $F=F_p$ a field with $p$ elements. $V$ is a vector space, $n$-dimensional over $F$. Calculate the number of one-dimensional vector spaces in $V$. I tried to solve it, but ...
3
votes
1answer
29 views

Field of rational functions

Let $K$ be a field with characteristic $p>0$ and $M=K(X,Y)$ the field of rational functions in 2 variables over $K$. We consider the subfield $L=K(X^p,Y^p)\subset M$. Show that $[M:L]=p^2$. I ...