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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4
votes
1answer
74 views

Find the field by the its multiplicative group

Suppose we have a group G. Is this a multiplicative (or additive) group of some field? I think that аn arbitrary group is not suitable (e.g. in the case of finite fields multiplicative group should be ...
1
vote
4answers
39 views

Show $α^{ −1}$ is algebraic over $ F $ of degree $n$.

Let $E, F$ be distinct fields such that $E$ is a field extension of $F$. Show that if $\alpha \in E \setminus F$ is algebraic over $F$ of degree $n \in \{2, 3, \cdots\}$, then $α^{ −1}$ is ...
4
votes
4answers
111 views

$\mathbb{Q}\left ( \sqrt{2},\sqrt{3},\sqrt{5} \right )=\mathbb{Q}\left ( \sqrt{2}+\sqrt{3}+\sqrt{5} \right )$

Prove that $$\mathbb{Q}\left ( \sqrt{2},\sqrt{3},\sqrt{5} \right )=\mathbb{Q}\left ( \sqrt{2}+\sqrt{3}+\sqrt{5} \right )$$ I proved for two elements, ex, $\mathbb{Q}\left ( \sqrt{2},\sqrt{3}\right ...
3
votes
1answer
26 views

$f(x)$ irreducible in $F[x]$, $\alpha$ a root, show that if some odd degree term of $f(x)$ has nonzero coefficient then $F(\alpha)=F(\alpha^{2})$

Let $F$ be a field, $f(x)$ an irreducible polynomial in $F[x]$ and $\alpha$ a root of $f$ in some extension of $F$. Show that if some odd degree term of $f(x)$ has a nonzero coefficient, then ...
1
vote
0answers
37 views

Is it possible to define a continuous field with characteristic $\neq 0$?

For example, defining an addition and multiplication on the unit circle in the complex plane such that it forms a field. This would be a sort of continuous analog of the finite fields. Another way I ...
2
votes
2answers
37 views

Show that every $f(x) \in K[x]$ can be represented as $g(x^{p^e})$

A (probably simple) question I encountered but I don't know how to approach: Let $K$ be a field of prime characteristic $p>0$. Show every $f(x) \in K[x]$ can be represented as $g(x^{p^e})$ ...
3
votes
1answer
33 views

Linear combination of matrices over finite and infinite fields

Let $F\subset K$ be the fields. Let $A_1,\ldots, A_m$ be the $n\times n$ matrices over the field $F$, and $c_1,\ldots,c_m\in K$ such that $c_1A_1+\cdots+c_mA_m$ is invertible. How to prove that for ...
0
votes
2answers
47 views

Show $f$ is irreducible.

Let $E$ be an extension field of $F$. Show that if $\alpha \in E$ is algebraic of degree $n$ over $F$ and $f\in F[X]$ is of degree $n$ with $f(\alpha) = 0$, then $f$ is irreducible. For this ...
1
vote
1answer
46 views

Every finite abelian extension of Q contains a totally real subfield of index 2?

I can reduce this to the case of cyclotomic field extensions, by embedding the abelian extension into a cyclotomic extension and using the "sliding-up" lemma. I am stuck on how to prove this for the ...
0
votes
0answers
12 views

Every Transcendence Basis Has the Same Cardinality

I'm trying to understand Theorem 12.2 at this link: http://ocw.mit.edu/courses/mathematics/18-782-introduction-to-arithmetic-geometry-fall-2013/lecture-notes/MIT18_782F13_lec12.pdf. In particular, I ...
5
votes
1answer
72 views

Show $x^p-t$ has no root in the field $\mathbb{F}_p(t)$

I don't think I fully understand. Let's say there is a root $x_0 \in K=\mathbb{F}_p(t)$, where $p$ is a prime number. Then $x_0 = \frac{P(t)}{Q(t)}$ for some polynomials $P,Q \in \mathbb{F}_p[t]$. ...
-5
votes
2answers
49 views

Fields in Abstract Algebra [duplicate]

How to prove the following: Show that $\mathbb Z_{n}$ is a field if and only if $n$ is prime.
1
vote
1answer
25 views

the degree of every irreducible polynomial that divides $x^p-x-a$ is the same.

let $F$ be a field with char$(F)=p>0$ where $p$ is a prime.given $a\in F^\times $ ($a\not=0$) denote \begin{equation*}f(x)=x^p-x-a\end{equation*} I'm trying to prove that the degree of every ...
0
votes
2answers
25 views

$\Bbb{Z}_{2}(\alpha)$ as splitting field

i have problems with an exercise: let $\alpha$ be a root of the polynomial $X^{3}+X^{2}+1$ in $\Bbb{Z}_{2}$. Prove that $\Bbb{Z}_{2}(\alpha)$ is the splitting field of this polynomial over ...
2
votes
2answers
26 views

If chatacteristic of $K$ is positive, show that every homomorphism from additive group to multiplicative group maps all elements of $K$ to $1$

Suppose $K$ is a field. Denote $(K,+)$ as the abelian group under addition operation and $(K,\times)$ as the abelian group under multiplication opearation. If the characteristic of $K$ is positive, ...
0
votes
2answers
54 views

Algebra and Maximal ideal.

I am trying to solve the following problem. If $ \mathcal{K}$ is a field and $a_1,a_2,\dots,a_n \in \mathcal{K}$. Prove that $(x_1-a_1,x_2-a_2,\dots,x_n-a_n)$ is a maximal ideal in ...
2
votes
1answer
25 views

Is $\alpha$ a norm in the extension $K(\sqrt[n]{\alpha})$?

I'm having trouble wrapping my head around this. $K$ is a field of characteristic zero containing all $n$th roots of unity, and $\alpha \in K$. Let $L = K(\sqrt[n]{\alpha})$, $\mu$ the minimal ...
4
votes
0answers
41 views

An infinite prime can ramify right? (So what is Neukirch talking about?)

I have been under the impression for several years that if $L/K$ is an extension of number fields, then an infinite place of $K$ is said to ramify in $L$ if it comes from a real embedding of $K$ which ...
1
vote
0answers
11 views

Why is an $n$th power a norm in a Kummer extension?

Let $F$ be a $p$-adic field containing the $n$th roots of unity. Then by Kummer theory, $[F^{\ast} : F^{\ast n}]$ (which is finite) is equal to the cardinality of $\textrm{Gal}(E/F)$, where $E$ is ...
3
votes
1answer
80 views

Matrix and field extension

It is given that $F\subset K$ are fields. $A$ is a matrix of size $n\times n$ over $K$. I need to prove that there exist $c_1,\ldots,c_k\in K$, linearly independent over $F$, and matrices ...
4
votes
3answers
83 views

Show that $\mathbb{Q}(\sqrt{2 +\sqrt{2}})$ is a cyclic quartic field i.e. is a galois extension of degree 4 with cyclic galois group

Show that $\mathbb{Q}(\sqrt{2 +\sqrt{2}})$ is a cyclic quartic field i.e. is a galois extension of degree 4 with cyclic galois group with some elementary algebra, $x - \sqrt{2 +\sqrt{2}} = 0 ...
2
votes
1answer
34 views

Existence of multiplicative inverse in field $\mathbb{Q}(\sqrt{d})$

Let $\mathbb{Q}(\sqrt{d}) = \{a + b \sqrt{d}: a,b \in \mathbb{Q} \}.$ Show that $\mathbb{Q}(\sqrt{d})$ is a field. Everything seems obvious except for existence of inverses in the multiplicative ...
2
votes
0answers
31 views

Transcendence Degree of Integral Domain over a Field

This may be trivial, but I am confused on the following issues. 1) If we have a finitely generated integral domain R over a field k, why is the transcendence degree of R over k (that is, the ...
0
votes
1answer
36 views

Radical extension with root of cubic polynomial

If I take $f(x)$ is an irreducible cubic over $\mathbb{Q}$ with a root $\alpha$ in a splitting field and given that $\mathbb{Q}(\alpha)$ is a radical extension is it true that $\mathbb{Q}(\alpha) = ...
0
votes
2answers
37 views

Show that $\mathbb{F}_q$ is a Splitting Field for Polynomial $x^q-x$

I have been given a homework problem which asks me to assume that $\mathbb{F}_q$ is a field of order $q$, and to show that $\mathbb{F}_q$ is a splitting field for the polynomial ...
0
votes
1answer
44 views

Prove that $L=K[x]/\langle m(x)\rangle$ is an algebraic extension of $K$

Let $m(x)$ be an irreducible polynomial of degree $n$. How can I prove that $$L=\frac{K[x]}{\langle m(x)\rangle}$$ is: A vector space of dimension $n$, An algebraic extension of $K$? Edit: I'm ...
0
votes
0answers
17 views

Is there a natural link between symmetric polynomials and symmetric algebra?

Let $R$ be a commutative ring and $R[X_1,...,X_n]^{S_n}$ be the ring of symmetric polynomials. I have learned some basic properties of this ring and the results are really similar to those by ...
0
votes
1answer
36 views

If you have a field isomorphism and the domain is algebraically closed then so is the image?

I know it makes sense because if they are isomorphic they are practically the same thing, but what would a proof look like?
1
vote
1answer
31 views

Quadratic Extensions

I am having a hard time understanding the concept of quadratic extensions. My book explains it: If the minimum polynomial of $a$ over a field $F$ has degree 2, we call $F(a)$ a quadratic ...
0
votes
2answers
18 views

What is an embedding of extensions?

I'm given a definition that I don't understand. I just want to have an understanding of it. It goes as follows. We have two Field extensions $H$ and $K$ of a field $F$ and a map $v: K \to H$. They ...
0
votes
2answers
38 views

Field Extension Question for Polynomials

I cannot seem to find the answer to this question on the internet. It is a question about field extensions for an element $a,b \neq F$ but in some extension $K$. I am wondering if $F(a,b)= \lbrace ...
3
votes
1answer
67 views

Fields of characteristic $p$ that are isomorphic as vector spaces, but not as fields

Give an example of a field $\mathbb{K}$ of characteristic $p > 0$ and elements $a,b \in \bar{\mathbb{K}}$, such that $\mathbb{K}(a)$ and $\mathbb{K}(b)$ are isomorphic as vector spaces over ...
5
votes
3answers
88 views

Find the Galois group of the polynomial $x^{4} + x - 1$ over the finite field $\mathbb{F}_{3}$

Find the Galois group of the polynomial $x^{4} + x - 1$ over the finite field $\mathbb{F}_{3}$. I'm particurly struggling with what would be the approach to finding the splitting field of this ...
4
votes
0answers
37 views

extending isomorphism to an automorphism

Lets say $K/F$ is a field extension and $\alpha ,\alpha '\in K$ are two distinct roots of the same irreducible polynomial in $F[x]$. there exists an isomorphism $$\psi:F(\alpha)\rightarrow ...
1
vote
0answers
29 views

Galois group of $ \mathbb{Q}(\varepsilon_5) / \mathbb{Q} $

I'm trying to solve the following problem: Let $ L =\mathbb{Q}(\varepsilon_5) $ be an extension of $ \mathbb{Q} $. Find the Galois group $ G(L / \mathbb{Q})$ $ \varepsilon_5 $ is the primitive root ...
1
vote
3answers
36 views

What is the number of elements in $Aut(Q(\pi)/Q)$?

I tried to prove that $|Aut(E/F)|$ is finite, then $E/F$ is a finite extension, but then now I think $Q(\pi)/Q$ would be a counterexample for this. I can see that there are two automorphisms ...
0
votes
2answers
63 views

Applications of $\mathbb{Z}/n\mathbb{Z}$ [closed]

I would like someone to proof me this claim and give me its applications in mathematics if it's not a convention. Claim: for all positive integers $n$, the ring $\mathbb{Z}/n\mathbb{Z}$ is domain if ...
3
votes
0answers
30 views

Does $[E:F]=|Aut(E/F)|$ imply Galois extension?

Let $E/F$ be a finite field extension such that $[E:F]=|Aut(E/F)|$. Then, is $E/F$ Galois? Even though I have proven it, I'm not sure of it. Is this really true? Here's how I proved it: Let $\bar ...
1
vote
1answer
37 views

Smallest field containing $F$ and $a \in K$

Definition. Given a field extension $K \supset F$ and an element $a \in K,$ define $F(a)$ to be the intersection of all subfields of $K$ that contain $F$ and $a.$ What is some more explicit notation ...
0
votes
3answers
61 views

Finding the fixed Field of $\sigma \in Aut(\mathbb{R}(t)/\mathbb{R})$

Finding the fixed Field of $\sigma \in Aut(\mathbb{R}(t)/\mathbb{R})$ Let $\sigma$ be such that $\sigma(t)=-t$. I assume there is only one automorphism like this, I am not sure exactly why... How ...
7
votes
4answers
205 views

Fields of arbitrary cardinality

So I took an introductory abstract algebra course a few semesters ago, and we were shown that groups and rings can both be made into products, i.e. if I have some group $G$ (resp. ring $R$) and some ...
0
votes
1answer
28 views

What is a Galois closure and Galois group?

I was reading wikipedia for Galois groups and this term suddenly appears and there is no definition for it. What is a Galois closure of a field $F$? Does this mean a maximal Galois extension of $F$ ...
3
votes
1answer
24 views

The necessary and sufficient condition for a regular n-gon to be constructible by ruler and compass.

I have a problem concerning the necessary and sufficient condition for a regular n-gon to be constructible by ruler and compass. $\bf My$ $\bf question:$ For a given positive integer $n$, how can we ...
4
votes
0answers
19 views

Why is this a corollary of this theorem?

Lang - Algebra p.251 Proposition 6.11 Let $E/F$ be a normal field extension. Let $E^G$ be the fixed field of $Aut(E/F)$. Then, $E^G$ is purely inseparable over $F$ and $E$ is separable over ...
1
vote
1answer
22 views

Can a field extension still have “non-separability” above its maximal purely inseparable subextension?

Question 1 Let $E/F$ be an algebraic field extension. Let $K$ be the set of all elements of $E$ that are purely inseparable over $F$. Then, $E/K/F$ is a tower of fields, and $K/F$ is purely ...
6
votes
0answers
60 views

Let $\mathbb{K} $ be a field of characteristic $p>0$ and $\mathbb{F} | \mathbb{K} $ a finite and separable extension.

Let $\mathbb{K}$ be a field of characteristic $p>0$ and $\mathbb{F}/ \mathbb{K}$ a finite and separable extension. Show that if $B=\{\alpha_1,\dots,\alpha_n\}$ is a basis, then ...
2
votes
1answer
83 views

Roots of $f(x) = x^3+x^2-2x-1$

Roots of $f(x) = x^3+x^2-2x-1$ Show: $a_1=2\cos(\frac{2\pi}{7})$ is a root of $f$. [Edited here] $a_2 = a_1^2-2$ is a root of $f$. $a_3 = a_1^3-3a_1$ is a root of $f$. The first one is ...
4
votes
0answers
51 views

Existence of Jordan decomposition over finite field

Prove that over finite field $\mathbb F$ exists additive Jordan-Chevalley decomposition: for all matrix $M$ there are semisimple matrix $M_{s}$ and nilpotent matrix $M_{n}$ such that $M=M_{s}+M_{n}$. ...
0
votes
0answers
20 views

What is a purely inseparable extension?

There are many different definitions of purely inseparable extension, and below is what I have chosen for my definition. (Since I don't know what is a standard one, if you know please tell me what ...
2
votes
2answers
63 views

Is the degree of an infinite algebraic extensions always countable?

I guess this is right and try to prove it by using the fact that the polynomial ring $K[t]$ has a countable basis $1,x,x^2,\cdots$. But How to use this fact? Aside, if this statement is true. Is the ...