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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1answer
40 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
31 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 ...
3
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0answers
42 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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0answers
12 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.
2
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1answer
30 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
29 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
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1answer
33 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
24 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 ...
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0answers
33 views

Why do I get all the $K$-automorphisms of a splitting field $K''$ successively?

Please let me explain my question on a specific example: Let $K=\mathbb{Q}$ and let $K''=\mathbb{Q}(\sqrt[3]2, \xi)$ be the splitting field of the polynomial $f=X^3-2\in K[X]$. The polynomial $f$ is ...
2
votes
1answer
55 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 ...
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vote
2answers
36 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
33 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 $K(a)=K(a^2)$.
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0answers
11 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
61 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
34 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 ...
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0answers
25 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 ...
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0answers
22 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
50 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
30 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
41 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 ...
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vote
2answers
41 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 ...
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vote
1answer
41 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
27 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 ...
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vote
0answers
44 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 $x^k=n\cdot a$, ...
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0answers
32 views

How to find field embeddings

What are the field embeddings from $K\to\bar{\mathbb Q}$ and how can I use these to compute $N_{K/\mathbb Q}(a)$ and $T_{K/\mathbb Q}(a)$, with $K=\mathbb Q({\sqrt[3]{2}})$ and $a={\sqrt[3]{2}}+3$ ...
0
votes
1answer
20 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
24 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
34 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 ...
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vote
2answers
66 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? ...
0
votes
0answers
21 views

Q[i]={a+bi is in C|a,b is in Q}. Prove that Q[i] is a subfield of C and Q[i] is isomorphic to field of quotients Z[i]

How would you do this? First off I'm not so sure what are the specific criteria we need to check for a subfield and how to show Q[i] is isomorphic to field of quotients. I know Q[i] can be shown as ...
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0answers
17 views

Criteria to check for subfield and subdomain

I've been doing many problems that go like here's a field verify this is a subfield or here's a domain, verify this is a subdomain. However its often very tricky for me to see it and write clear ...
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votes
1answer
26 views

Module and Noetherian/Artinian Rings

I am trying to prove that: Every finitely generated $F$-module $M$ is both Noetherian and Artinian where $F$ is a field. For this I am looking at the submodules of $F$ and saying that they are in ...
1
vote
1answer
38 views

If $a$ is algebraic over $F$ and $b$ is transcendental over $F$, then $a+b$ is transcendental over $F$.

Let $a$ and $b$ be elements in extension field $F$. Is it true that: If $a$ is algebraic over $F$ and $b$ is transcendental over $F$, then $a+b$ is transcendental over $F$? I just did the same ...
2
votes
1answer
29 views

Is the formula $[k(a,b):k][k(a) \cap k(b):k] = [k(a):k][k(b):k]$ true for simple algebraic extensions ($k(a)/k$ and $k(b)/k$ )?

Let $k$ be any field and $L/k$ be a field extension. Suppose $a, b \in L$ are algebraic over $k$. Is the formula $[k(a,b):k][k(a) \cap k(b):k] = [k(a):k][k(b):k]$ true? This formula comes from page ...
2
votes
1answer
41 views

Finite separable field extensions such that $KL/L$ and $K/K\cap L$ have non-isomorphic automorphism groups

If $K/F$, $L/F$ are finite separable extensions (not necessarily finite Galois extensions), then it seems clear that $KL/F$ is also a finite separable extension. However, in this case, is ...
5
votes
1answer
79 views

Why is the extension $k(x,\sqrt{1-x^2})/k$ purely transcendental?

Consider the function field $k(x,\sqrt{1-x^2})$ of the circle over an algebraically closed field $k$. Is $k(x,\sqrt{1-x^2})$ a purely transcendental extension over $k$? I'm curious because I was ...
1
vote
1answer
83 views

If $a$ is a complex number s.t. $a\notin \mathbb R$, then $\mathbb R(a)=\mathbb C$?

If $a$ is a complex number s.t. $a\notin \mathbb R$, then $\mathbb R(a)=\mathbb C$? I'm asked to give a proof or a counterexample. I'm a bit confused on the notation of $\mathbb R(a)$, what does this ...
3
votes
1answer
37 views

Find the number of integers $r$ such that the polynomial $x^{r}-a$ has a linear factor over $\mathbb{F}_{p^{n}}$

If we have a finite field $\mathbb{F}_{p^{n}}$, how does one determine the number of integers $r$ in $\{0,1, \ldots, p^{n}-2 \}$ for which the equation: $x^{r}=a$ has a solution for every $a \in ...
0
votes
1answer
13 views

Prime subfield equivalent definitions

So I have field $F$ (any characteristic), and its prime subfield $K$. I have three definitions: (i) that $K$ is the subfield of $F$ such that $K$ has no proper subfield; (ii) that $K=\bigcap_i K_i$ ...
2
votes
1answer
55 views

Which Galois Field is isomorphic to this extension?

Let $\alpha$ be an element in an algebraic closure of $GF(64)$ such that $\alpha^4=\alpha+1$. For which $r\in \mathbb{N}$ is $GF(64)$ adjoined $\alpha$ isomorphic to $GF(2^r)$? [Adding the following ...
0
votes
1answer
26 views

Reducing a sum of products of primitive nth roots

QUESTION: Suppose $\zeta$ is an primitive n-th root of unity. And suppose $n=p^r$ where $r\geq 1$. What is $(1-\zeta)\zeta^{p^r-p^{r-1}-1}$ written as the sum of the basis elements ...
5
votes
3answers
94 views

A good introductory book on Ring and Field theory with a view towards Number Theory ?

Please suggest some good introductory books on Rings&Fields with a view towards Number Theory ?
2
votes
1answer
50 views

What happens with $S_n$ in rings, integral domains and fields?

From Cayley's theorem we know that every group is a symmetric group, i.e. a group of permutations. But what happens when we "extend" a group to a ring or a field for example; is there any ...
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2answers
68 views

Showing a counterexample regarding normal extension

For field extensions K/E, E/F, if K/F is a normal extension, E/F is a normal extension also? I think this is false..but can't find a counterexample. Could anyone suggest me some example?
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0answers
11 views

Verification of proof that the left distributive property holds on a field of quotients, F

Here's my proof. I would like to know whether it is right or wrong and if the latter where it needs to be corrected: (a,b)=[(a,b)][(cf+ed,df)]=[(acf+aed,bdf)]. We want to show its equivalence to: ...
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0answers
30 views

the motivation of separable field extension

What is the origin of the motivation of separable field extension? Is it related to separable topological space or something else?
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3answers
91 views

If $F$ is an extension field of the field $K$ such that $[F:K] =1$, then $F=K$

Suppose $F$ is an extension field of the field $K$ such that $[F:K] =1$. How to prove that $F=K$? Thank you for your time and help.
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0answers
55 views

Isomorphism for the group of units of the ring of integers of a local field

Let $K$ be a local field with a discrete and non-archimedean absolute value, $\mathcal{O}_K$ be its ring of integers, $\mathfrak{m}_K$ be the unique maximal ideal of $\mathcal{O}_K$ and ...
0
votes
1answer
18 views

Explaining a. Q(R,T) has unity even if R does not and b. In Q(R,T) every nonzero element of T is a unit

So I've been having trouble understanding this proof for quite a while now. I understand how the field of quotients is formed but not so much of why my professor's answers use this tactic for this ...