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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2
votes
0answers
21 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:   $[F(x+y):F] \mid [F(x,y):F]$   ...
2
votes
2answers
45 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 ...
1
vote
0answers
36 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 ...
1
vote
0answers
19 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 ...
-5
votes
0answers
53 views

Give an example of a field K, an extension field F, a subring R of F containing K where R is not a field? [on hold]

Give an example of a field K, an extension field F, a subring R of F containing K where R is not a field? I had $\mathbb{C}$ as a field, $\mathbb{C}(x)$ as a field extension, and $\mathbb{C}[x]$ ...
2
votes
1answer
81 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 ...
0
votes
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 ...
0
votes
2answers
28 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\\ ...
4
votes
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 ...
0
votes
1answer
15 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 $F$. ...
0
votes
0answers
13 views

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 $n \geq 1$. ...
0
votes
1answer
20 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 ...
1
vote
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 $L_1 \cap L_2$?
1
vote
1answer
43 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 ...
0
votes
0answers
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 ...
1
vote
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
votes
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??
0
votes
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 $\{\beta_1,...,\beta_n\}$ ...
2
votes
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?)
5
votes
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
votes
1answer
18 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 ...
1
vote
2answers
56 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
votes
1answer
34 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 ...
0
votes
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 $\phi_n=e^\frac{2\pi i}{n}$ .
0
votes
1answer
13 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 ...
3
votes
0answers
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 ...
0
votes
0answers
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.
1
vote
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 $\Bbb Q$.
0
votes
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 ...
4
votes
1answer
22 views

Factor $x^2+2x+2$ in $\mathbb{F}_3/(x^2+1)$

I am asked to find two roots of $x^2+2x+2$ in $\mathbb{F}_3[x]/(x^2+1)$ (the Kronecker construction). The elements of that field are (equivalence classes of) constant or linear polynomials in ...
2
votes
1answer
46 views

Galois group of $x^{15}-1$

Let $\zeta$, $\eta$, $\omega$ denote the primitive fifteenth, fifth, and cube roots of unity. a) Describe all the automorphisms in $G=G(\mathbb Q (\zeta)/ \mathbb Q)$. b) Show that $G$ is isomorphic ...
1
vote
2answers
31 views

Find the field of intersection

Let $\mathbb{F}_{p^{m}}$ and $\mathbb{F}_{p^{n}}$ be subfields of $\overline{Z}_p$ with $p^{m}$ and $p^{n}$ elements respectively. Find the field $\mathbb{F}_{p^{m}} \cap \mathbb{F}_{p^{n}}$. Could ...
1
vote
1answer
27 views

Let $F|K$ be a field extension and $a \in F $ such that $[K(a):K]$ is odd integer [duplicate]

Let $F|K$ be a field extension and $a \in F$ such that $[K(a):K]$ is odd integer, then prove that $K(a)=K(a^2)$.
0
votes
0answers
7 views

Purely inseparable extension from Hungerford

Hungerford, Algebra, V.6.4 says, $F/K$ is purely inseparable if and only if $F$ is generated by a set of purely inseparable elements over $K$. My question : is there any purely inseparable extension ...
1
vote
1answer
57 views

Proving irreducibility of polynomial over various fields

I am working on a problem set from an online course and am struggling to understand a key concept related to proving reducibility / split-ability etc. in a finite field $\mathbb{F}_{p^n}$, as well as ...
0
votes
1answer
31 views

Field algebraically closed

The problem is Let $E$ a finite extension of $F$ and $E$ is algebraically closed, show that $F$ is perfect. I know that a field $F$ is called perfect if every irreducible polynomial in $F[x]$ is ...
1
vote
0answers
21 views

Euler's criterion and Legendre symbol

I am working on an exercise which is the following : Let be $n$ an odd integer and $b$ such as $b \wedge n = 1$, then $(\frac{b}{n}) \equiv b^{(n-1)/2} \mod n$. (*) If $n$ is divisible by the ...
0
votes
0answers
21 views

Show that a heptagon is not constructible with a compass and straight edge.

I've so far worked out that we need to prove that we cant construct the angle $\frac{2\pi}7$, or that point on the unit circle $e^{i\frac{2\pi}7}$. What I dont understand is solving for the roots of ...
5
votes
1answer
44 views

Decompose $a = a_1\cdots a_k$ and $a_1 + \dots +a_k = 0$

Problem. Prove that in the field $F, \text{char }F\neq2$ every element $a$ can be decomposed in the following way: $a = a_1\cdots a_k$ and $a_1 + \dots +a_k = 0$. Attempt 1. For the fields in which ...
2
votes
1answer
27 views

Proof of an existence of an algebraic closure of a given field

I am studying the field theory with Abstract Algebra by Dummit & Foote. A proof of an existence of an algebraic closure by constructing such extension using the Zorn's lemma is given in this ...
0
votes
1answer
40 views

Linear transformations of fields

Let $K$ be field. Show if $f: K \to K$ is linear transformation, then there exists $a \in K$ such that for every $x \in K$, $f(x) = ax$ I don't know how to prove it but for instance $a = 1$ satisfy ...
1
vote
2answers
36 views

Why two extension fields are isomorphic as vector spaces but not fields?

I understand that $\mathbb{Q}(\sqrt{2})$ and $\mathbb{Q}(\sqrt{3})$ are isomorphic as vector spaces but not as fields. However, I do not understand why that is true. What is happening when they are ...
0
votes
1answer
32 views

Good book for Local Fields/ Commutative algebra?

I am currently studying Local Fields from Serre's textbook, but finding that it requires a bit too much prior knowledge for me. Can anyone suggest another book that I can use alongside Serre that ...
0
votes
1answer
24 views

Why the ideals here are in this form?

In this article, in the proof of problem no. 6, p. 3, it listed all the possible ideals, because they contains at least one of $2$, $3$, $5$, from $120$. And at least one of $x+1$, $x^2-x+1$. But I ...
1
vote
0answers
43 views

$x^k = na$ has solution over the field $F, \text{char} F \neq 2$

I came up with an interesting Hypothesis. Suppose we are in the field $F, \text{char }F \neq 2$. Let's fix an arbitrary element $a \in F$. Is it true that at least one equation $x^k=n\cdot a$, ...
0
votes
1answer
17 views

How to show $[F(a):F[a^4+a^2+1]$ is finite.

I have an element $a$ in an extension field $F$. I'm asked if it is true that $a^4+a^2+1$ is algebraic over $F$ if and only if $a$ is algebraic over $F$. I know that if I can show ...
2
votes
1answer
22 views

$K$ is an extension of $F$ and $a,b\in K-F$. If $ab$ is algebraic over $F$ then $b$ is algebraic over $F(a)$.

Let $F$ be a field and $K$ be an extension field of $F$. Proof\Counterexample: $K$ is an extension of $F$ and $a,b\in K-F$. If $ab$ is algebraic over $F$ then $b$ is algebraic over $F(a)$. I haven't ...
0
votes
1answer
32 views

Quadratic extension of quadratic extensions

I need help for the following exercise: The field $\mathbb{Q}(e^{\frac{2 \pi i}{3}})$ is a quadratic extension of $\mathbb{Q}$ and $\mathbb{Q}(e^{\frac{\pi i}{6}})$ is a quadratic extension of ...
1
vote
2answers
61 views

Can $\mathbb{Z}_{6}$ be a subring of some field?

I think the answer is yes because $\,\mathbb Z_{6}\,$ is a ring, it has a unity and has multiplicative inverse and its elements are commutative so it can be a subring of a field. Is this correct? ...