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
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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 ...
0
votes
0answers
20 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 ...
4
votes
1answer
72 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 ...
0
votes
1answer
21 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 ...
2
votes
1answer
68 views

Which one is a field?

Which one is a field? i) $\cfrac{\mathbb{Z}[x]}{\langle{x^2+2}\rangle}$ ii)$\cfrac{\mathbb{Q}[x]}{\langle{x^2-2}\rangle}$ I think both are correct because for both the cases ...
0
votes
0answers
14 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 ...
1
vote
1answer
81 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 ...
1
vote
1answer
31 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
26 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
39 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 ...
3
votes
1answer
22 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 ...
2
votes
1answer
53 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
12 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 $K=\bigcap_i K_i$ ...
0
votes
1answer
23 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 ...
2
votes
1answer
46 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 ...
0
votes
0answers
27 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?
1
vote
2answers
66 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?
3
votes
3answers
89 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.
1
vote
0answers
9 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: ...
4
votes
3answers
3k views

What is a primitive polynomial?

What is a primitive polynomial? I was looking into some random number generation algorithms and 'primitive polynomial' came up a sufficient number of times that I decided to look into it in more ...
1
vote
0answers
48 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
13 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 ...
4
votes
0answers
41 views

Is there a recurrence formula for finding primitive polynomials in order to construct $\mathbb{F}_{p^n}$?

(In a previous question, I asked some related question but it was very disorientingly formulated. I apologize for that.) Let $p$ be a prime and $n\geq 1$ an integer. The finite field ...
0
votes
0answers
21 views

Embedding of splitting field for a family of polynomials

STATEMENT: Let $K$ be a splitting field for the family $\left\{f_i\right\}_{i\in I}$ and let $E$ be another splitting field. Any embedding of $E$ into $K^a$ inducing the identity on $k$ gives an ...
1
vote
1answer
14 views

separable polynomial

How to show that if $K$ is a field of characteristic $p$ with $p$ prime and if $f(X)\in K[X]$ is an irreducible and inseparable polynomials, therefore there exist a $d\in\mathbb N, d>0$ such ...
0
votes
1answer
26 views

Verification of proof that for a,b in ring R, assuming ab is a zero divisor at least one of a and b is zero divisor

I'm not so sure if this is correct but here's what I have so far: ab is a zero divisor iff there is a c$\neq$0 s.t. (ab)c=c(ab)=0 given ab$\neq$0 and c$\neq$0. Then we have ...
7
votes
1answer
2k views

Existence of irreducible polynomials over finite field

Let $F$ be a finite field. How do we prove that for each $n \in \mathbb{N}$ there is an irreducible polynomial of degree $n$? One can assume that $F = \mathbb{F}_{p^m}$ where $p$ is prime. If $n \ge ...
1
vote
0answers
35 views

Degree of the field extension for $x^4+3$

I want to find a splitting field for : $x^4+3 \in \mathbb{Q}[x]$ and the degree of that extension , I would be thankful if you verify/correct what I have tried for this problem : First I find the ...
0
votes
3answers
67 views

Why is $\alpha$ transcendental over $K$

Let $K$ be a field and let $\alpha$ be an element of the field $K(T)$ of rational functions, with $\alpha\not\in K$. Prove that $\alpha$ is transcendental over $K$. In this case $\alpha$ is of ...
2
votes
2answers
35 views

show that additive group of field of characteristic 0 is not cyclic

Show that additive group of field of characteristic 0 is not cyclic. If it is so then the additive group will be isomorphic to $\Bbb Z$ from here how do I proceed. I have seen Why must a field with ...
6
votes
1answer
90 views

Subfield Criteria - Proof or Counterexample

I am interested in whether the following claim is true for all fields $F$: Conjecture: A subset $X\subset F$ is a subfield if and only if (1) $1\in X$, (2) $x,y\in X\Rightarrow x-y\in X$; and (3) ...
1
vote
3answers
73 views

Calculate the dimension of the field extension $[\mathbb{Q}[ \sqrt2 , \sqrt3] : \mathbb{Q}]$

I've though that $[\mathbb{Q}[ \sqrt2 , \sqrt3] : \mathbb{Q}] = [\mathbb{Q}[ \sqrt2] : \mathbb{Q}].[ \mathbb{Q}[\sqrt2, \sqrt3]:\mathbb{Q}[ \sqrt2] ] $ And I know how to prove $[\mathbb{Q}[ \sqrt2] : ...
0
votes
0answers
29 views

Example of a local field of positive characteristic?

I am looking for a local field of positive characteristic, like $Q^{2}_{2}$ was used in this article: in fact, i need an another Example of a local field of positive characteristic like $Q^{2}_{2}$ ...
2
votes
1answer
35 views

More general constructible numbers?

I've recently learned about the field of constructible numbers (those which can be constructed with compass and straight-edge). A theorem in this subject states that a number $z$ (real or complex) is ...
3
votes
1answer
47 views

$x^{p^2-1}-1$ is divisible by $x^8-1$ when $p$ is odd?

In the proof of reducibility of $x^4+1$ over $F_p$ (which is stated as a corollary of the structure theorem of the finite field $F_{p^n}$), the following implication is used in the Algebra by Dummit ...
1
vote
2answers
29 views

Which general methods of field construction do we know?

This question is partially motivated by: http://mathoverflow.net/questions/3003/in-what-sense-are-fields-an-algebraic-theory whereas I am not a specialist in field theory. Q1: Which methods do we ...
3
votes
1answer
65 views

Prove that any subfield of $\mathbb C$ must contain $\mathbb Q$

I just started reading Linear Algebra by Hoffman and Kunze, and I came across the following line: The interested reader should verify that any subfield of $\mathbb C$ must contain every ...
1
vote
1answer
27 views

What exactly is $k\left(T_{n}\right)_{n\in\mathbb{N}}$?

Let $k$ be a field and $T_{n}$ indeterminates over $k$. Is $k\left(T_{n}\right)_{n\in\mathbb{N}}$ the field of fractions of the form $x=\frac{p}{q}$, where $p\in k\left[T_{i}\right]_{i\in\mathbb{N}}$ ...
0
votes
1answer
22 views

Algebraically closed fields minimal

STATEMENT: This is a portion from Lang's proof of theorem 2.8 in chapter V section 2. If $E$ is algebraically closed, and $L$ is algebraic over $\sigma k$, then $\sigma E$ is algebraically closed and ...
1
vote
3answers
38 views

Minimal Polynomial of $a +b\sqrt{2}$ as a function of a, b ∈ $\mathbb Q$

Determine the minimal polynomial over $\mathbb Q$ of $a +b\sqrt{2}$ as a function of a, b ∈ $\mathbb Q$. Let $x=a+b\sqrt{2}$ If $b=0$ then the minimal polynomial is $x-a$ if not, then ...
1
vote
0answers
25 views

Trivial absolute value

Let $K/L$ be a algebraic extension. Suppose that $\left|\cdot\right|$ is a absolute value in $K$ such that is trivially in $L$. Then is trivially in $K$. Thanks for anny suggestion. If is trivially ...
1
vote
1answer
26 views

Finite field extensions and minimal polynomial

I want to show the following statement: Let L/K be a finite field extension with $[L:K]=p$ for a prime $p$ Show that $[L:K]$ is simple Proof: 1) Choose $\alpha\in L$ with $\alpha \notin K$. Then ...
4
votes
5answers
135 views

Determine the minimal polynomial of $\sqrt 3+\sqrt 5$

I am struggling in finding the minimal polynomial of $\sqrt{3}+\sqrt{5}\in \mathbb C$ over $\mathbb Q$ Any ideas? I tried to consider its square but it did not helped..
-4
votes
1answer
219 views

Minimal polynomial over the field $\Bbb Q$

Compute the minimal polynomials over the field $\mathbb{Q}$ of the given numbers $\sqrt{2+i\sqrt{2}}$ $\sqrt{1+ \sqrt{3}}$ $5^\frac{1}{4}$
0
votes
1answer
25 views

Splitting field of polynomial

We consider the polynomial $f(x)=x^2+1 \in \mathbb{Z}_7[x]$ and let $E \subseteq \overline{\mathbb{Z}}_7$ be the splitting field. Let $F \subseteq \overline{\mathbb{Z}}_7$ the splitting field of the ...
0
votes
0answers
29 views

Degree of extension

Let $a, b \in \mathbb{C}$ and let $[\mathbb{Q}(a) : \mathbb{Q}]=m$, $[\mathbb{Q}(b) : \mathbb{Q}]=n$. Show that $[\mathbb{Q}(a,b):\mathbb{Q}]\leq mn$. If $(m,n)=1$ show that ...
0
votes
1answer
37 views

Show that $p=2^k+1$

When $p$ is an odd prime and $a=Re \left ( e^{\frac{2 \pi i}{p}} \right)$ then $[\mathbb{Q}(a) : \mathbb{Q}]=\frac{p-1}{2}$. Let $\theta = \frac{2 \pi}{p}$. If $\sin{\theta}$ is a constructable ...
2
votes
1answer
36 views

Irreducible polynomial/Splitting field

Let $f(x)=x^4+16 \in \mathbb{Q}[x]$. Split $f(x)$ into a product of first degree polynomials in $\mathbb{C}[x]$. Show that $f(x)$ is an irreducible polynomial of $\mathbb{Q}[x]$. Find the splitting ...
1
vote
1answer
19 views

Show that the equation has exactly $m$ different roots in the algebraic closure


Let $n=p^rm$, where $p$ is a prime, $m \in \mathbb{N}, r \geq 0$ an integer and $(p,m)=1$. 
I have to show that the equation $x^n=1$ has exactly $m$ different roots in the algebraic closure ...