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 ...

learn more… | top users | synonyms (1)

0
votes
1answer
36 views

Is a field just a commutative ring? [duplicate]

Is a field just a commutative ring? My algebra professor didn't give a very wide introduction to this algebraic structure, and I did not get a real grasp of what a field is. We're studying ...
2
votes
1answer
37 views

Multiple roots in $\mathbb{Z}_p$

Let f(x) ∈ $\mathbb{Z}$[x], a polynomial of degree n. Suppose f(x) has n distinct roots $a_1, ..., a_n$ ∈ $\mathbb{C}$. Now, with a given f(x), we call a prime p "bad" if f(x) has a ...
1
vote
0answers
28 views

Extension Field of $\mathbb{Q}$ and its Galois group

How many elements are in $Gal(\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5}:\mathbb{Q})$?
2
votes
0answers
31 views

Subgroups of Galois group and intermediate fields lattice for $(x^3-2)(x^2-3)$

I am trying to systematically determine all subgroups of Galois group and intermediate fields for $(x^3-2)(x^2-3)$(over $\mathbb Q$). It's not hard to determine the Galois group of $(x^3-2)$ and ...
1
vote
0answers
33 views

About a field extension and its normal closure

Is the extension $K=\mathbb Q(\root5\of2)$ over the rationals normal? If not find its normal closure. I know that K is not normal but I can not show and I think normal closure of $K$ is ...
2
votes
0answers
21 views

On transcendence base and separability

This is a problem in Hungerford's Algebra. Let $k$ be a perfect field and $F$ an extension field of $k$ with transcendence degree 1 and $F$ is not perfect. We have to show that $F$ is separably ...
0
votes
1answer
30 views

Finite fields and cardinality

I am trying to get my head around the proof of the following: Suppose K is a finite field. With $p=charK, |K|=p^r$ where r is a positive integer. I am supplied with the following proof: I do not ...
0
votes
3answers
47 views

Find $a\in\mathbb{R}$ such that $x_{1,2,3}\in\mathbb{Z}$

Consider $a\in\mathbb{R}$ and $x^3-x+a=0$ with $x_{1,2,3}\in\mathbb{C}$. We need to find $a\in\mathbb{R}$ such that $x_{1,2,3}\in\mathbb{Z}$. It seems be equivalent with to find a such that ...
1
vote
0answers
24 views

Linear recursions in finite fields

Let $F$ be a finite field and let $\alpha$, $\beta$ be distinct nonzero elements of $F$. Let $\alpha$ have order $r$ and let $\beta$ have order $s$. Let $M = \operatorname{lcm}(r, s)$. Let $a,b$ be ...
1
vote
1answer
21 views

Prove that the map $\theta(f(\alpha))=f^\sigma(\beta)$ is injective

I'm reading Lang's algebra chapter about Field theory and Galois theory. There is a theorem that says: Let $k$ be a field, $E$ an algebraic extension of $k$, and $\sigma:k\to L$ an embedding ...
0
votes
2answers
40 views

Find $\alpha$ s.t. $\mathbb{Q}(i,\sqrt[3]{2})$ is $\mathbb{Q}(\alpha)$ [closed]

I want to find $\alpha$ s.t. $\mathbb{Q}(i,\sqrt[3]{2})$ is $\mathbb{Q}(\alpha)$, but i'm not sure how to do that. $i^2 \in \mathbb{Q}$ and $\sqrt(3)^3 \in \mathbb{Q}$ and $2$ and $3$ are coprime ...
0
votes
1answer
18 views

separable polynomial $ \bmod p$ (definition)

Given a polynomial $ f(x) \in K[x]$, where $K$ is a number field, we say that $f$ is separable if all its roots are distinct in an algebraic closure of $K$. Question: What does it mean a ...
2
votes
2answers
38 views

Transcendence base of $\mathbb{C}$ over $\mathbb{Q}$ has infinitely many elements.

I need to show that transcendence base of $\mathbb{C}$ over $\mathbb{Q}$ has infinitely many elements. Since I do not know much about ordinals and cardinals, a proof based on algebra (rather than ...
-3
votes
2answers
40 views

prove $(a*b)\diamond (a*-b)=(a\diamond a)*(-(b \diamond b)$ [closed]

Let $F$ be a field under the operations ∗ and ◇ and for any element $x\in F$ we set $−x$ to be the (unique) element so that $x∗−x = −x∗x = e_∗.$ Use the field axioms to prove that for any two elements ...
5
votes
3answers
77 views

is a number field by definition a subfield of $ \mathbb C $?

I have seen that some authors are defing the number field as a subfield of $ \mathbb C$ which is a finite extension of the rational numbers $ \mathbb Q $, while some others without referering to ...
2
votes
2answers
23 views

behavior of a rational prime in quadratic extension (definition)

Let $ \mathbb Q \subset K=\mathbb Q (\sqrt{-n}) \subset L $, where $K/ \mathbb Q $ is a finite extension (i.e. $K$ is a number field) and $L/K$ is a maximal uramified abelian extension. If $p ...
1
vote
2answers
24 views

Different two definitions for separable extension

Let $E/F$ be an algebraic field extension and $\bar F$ be an algebraic closure of $F$. Define $[E:F]_{\text{sep}}$ as the cardinality of $$\{\sigma\in \operatorname{Mono}(E,\bar F): \sigma \text{ ...
0
votes
1answer
34 views

Hints on how to approach a problem concerning rings/field in Abstract Algebra

I am a student, prepping for a final exam in graduate Abstract Algebra. My professor has told me that he will be giving us the following two problems in class to turn in: (1) Given that R is an ...
1
vote
1answer
18 views

a simple example of crossed homomorphism for the proof of Hilbert Theorem 90

For a field extension $K/F$ and a subgroup $G$ of $Aut(K)$, A crossed homomorphism is defined to be a function $f:G \rightarrow K^*$ satisfying $$f(\sigma\tau)= f(\sigma)\cdot \sigma(f(\tau)) $$ I ...
0
votes
0answers
28 views

Splitting field of $f(x)=x^4+3$ in $\mathbb{Q}[x]$

I am trying to find the splitting field of $f(x)=x^4+3$ over $\mathbb Q$. It is irreducible, and the roots are ...
2
votes
2answers
45 views

Field extension of fixed field has degree greater than the size of the group

Let $K$ be a field, $G\leqslant\mathrm{Aut}(K)$ a (finite) group, and $K^G$ the fixed subfield of $K$. How exactly would you go about proving the following?$$[K:K^G]\geqslant|G|$$ For some reason I ...
5
votes
1answer
38 views

Show that $α$ is a perfect square in the quadratic integers in $\mathbb Q[\sqrt{d}]$

Question: Suppose that $\mathbb Q[\sqrt{d}]$ is a UFD, and $α$ is an integer in $\mathbb Q[\sqrt{d}]$ so that $α$ and $\barα$ have no common factor, but $N(α)$ is a perfect square in $\mathbb Z$. How ...
0
votes
0answers
25 views

Find the number of unitary group

Let K be a field of 25 elements. What is the number of 2 by 2 unitary matrix? My answer is 720 but I found solution by brutal force. Is there any nice method to calculate? Definition of Unitary ...
1
vote
1answer
77 views

The ideal $(p)$ always factors in the ring of integers of $\mathbb Q(\sqrt{2},\sqrt{3})$

Let $p$ be a prime integer. Is there a relatively elementary way to see that $(p)$ is never prime in the ring of integers of $\mathbb Q(\sqrt{2},\sqrt{3})$? One can prove this by looking at the ...
2
votes
0answers
67 views

Prove $λ$ must divide one of $x$, $y$, or $z$.

Let $λ = (3 + \sqrt{−3})/2 \in \Bbb Q[\sqrt{−3}]$. Prove that if $x^3 + y^3 = z^3$, and $x$, $y$, $z$ are quadratic integers in $\mathbb Q[\sqrt{−3}]$, then $λ$ must divide one of $x$, $y$, or $z$. ...
-1
votes
1answer
28 views

What does it mean for a field to be of order n?

I understand what a field is. But I do not understand what it means for a field to be of order n. Can someone explain please?
2
votes
2answers
40 views

if $P$ is a prime ideal of $O_K$, then $O_K/P$ is finite

let $P$ be a non-zero prime ideal of $O_K$, where $K$ is a number field(i.e. the degree $[K:\mathbb{Q}]$ is finite) then $O_K/P$ is finite. I'm working through a proof for this claim, however there is ...
1
vote
1answer
19 views

$O_k=\mathbb{Z}[\sqrt{d}]$, whenever $K=\mathbb{Q}(\sqrt{d})$ and $d\neq 1$ mod $4$

I'm going through a proof in my lecture notes for the mentioned statement. Showing $\mathbb{Z}[\sqrt{d}]\subseteq O_K$ was easy to understand, but then there's a few gaps when showing that ...
0
votes
0answers
25 views

Prove that if $x^3 + y^3 = z^3$, and $x$, $y$, $z$ are quadratic integers in $\mathbb Q[\sqrt{−3}]$, then $λ$ must divide one of $x$, $y$, or $z$.

Let $λ = (3 + \sqrt{−3})/2 ∈ Q[\sqrt{−3}]$. Prove that if $x^3 + y^3 = z^3$, and $x$, $y$, $z$ are quadratic integers in $\mathbb Q[\sqrt{−3}]$, then $λ$ must divide one of $x$, $y$, or $z$.
3
votes
2answers
45 views

How to show that $\mathbb{Q}(\sqrt{p},\sqrt{q}) \subseteq \mathbb{Q}(\sqrt{p}+\sqrt{q})$

How can I show that $\mathbb{Q}(\sqrt{p},\sqrt{q}) \subseteq \mathbb{Q}(\sqrt{p}+\sqrt{q})$, for distinct primes $p,q?$ The other inclusion is trivial. I tried saying $$(\sqrt{p}+\sqrt{q})^{-1} = ...
1
vote
0answers
19 views

Apply the Euclidean algorithm to $2$ and $1 − 3i$ in the integers of $\mathbb Q[\sqrt {-1}]$

Question: I have to apply the Euclidean algorithm to $2$ and $1 − 3i$ in the integers of $\mathbb Q[\sqrt {-1}]$ My Solution: Since $N(2) = 4$ and $N(1–3i) = 10,$ we must start by dividing $1–3i$ ...
1
vote
1answer
20 views

Find GCD in Q[√3] assuming it is defined

How do I find the GCD of 24 and 49 in the integers of Q[√3], assuming that the GCD is defined?
0
votes
1answer
25 views

Showing $\zeta_5 \notin \mathbb{Q}(\zeta_7)$

I was assigned this problem as homework, and got it wrong. I have not gotten a chance to ask the teacher about the solution. Can someone tell me why I am wrong, and how to do this correctly? Let ...
0
votes
1answer
28 views

Let ideal $I$ be generated by polynomial $p(x)$ in the ring $F[x]$ F-field.

Let ideal $I$ be generated by polynomial $p(x)$ in the ring $F[x]$ F-field. Prove: that the polynomials f(x), g(x) are in the same factor class of the ring $\implies f(x)=g(x)(mod\ p(x))$ ...
0
votes
2answers
35 views

Simple algebraic field extensions

Let $u$ be algebraic over a field $F$ such that $[F(u): F] = n$, and let $m$ be a natural number such that $(n,m!)=1$. Prove $F(u) = F(u^m)$. I know that since $u$ is algebraic over $F$, then $u$ ...
2
votes
4answers
52 views

The complex roots of a biquadratic polynom

In my recent post I have a problem with the following function: $x^4-4x^2+16$, and what I need is to find the complex roots. Here is my answer: First step, I make the substitution $x^2=y$ which ...
2
votes
2answers
61 views

Deriving a contradiction

How can I derive a contradiction from the following nasty statement: Assume $\sqrt{5} = a + b\sqrt[4]{2} + c\sqrt[4]{4} + d\sqrt[4]{8},$ with $a,b,c,d \in \mathbb{Q}$? This is the last piece of an ...
0
votes
0answers
34 views

Primitive element of finite field

I am looking for primitive element of galois field of order $8.$ So, I can look at the field $\mathbb{F}_8=\mathbb{Z}_2[x]/(x^3-x-1)$. I computed $\mathbb{F}_8^{\times}$ and now the primitive ...
0
votes
1answer
25 views

Characterization of separable elements in a field extension

Let $k,F$ be two fields with $char\ k = p >0$. Prove that an algebraic element $u \in F$ is separable over $k$ iff $k(u) = k(u^{p^n})$ $\forall n \in \mathbb{N}$. Again, still studying for my ...
1
vote
1answer
30 views

Weakly normal polynomials and normal polynomials.

I have been going through the notes of Prof. Pete L. Clark here (warning: long pdf). They are rough notes on Field theory and on page 30 he defines $P \in K[t]$ a normal polynomial if $P \in L[t]$ is ...
3
votes
1answer
60 views

Irreducible polynomial over a field $k$ with $char\ k = p > 0$

I'm studying for my Abstract Algebra II final and reviewing problems. I'm having some trouble with this one. Direction would be helpful. Let $k$ be a field with $char\ k = p > 0$, and let $f(x) ...
2
votes
1answer
28 views

Construction of the discrete valuation ring

Let $K$ be a field. A surjective transformation: $v: K \to \mathbb{Z}\cup\{\infty\}$ is defined as a discrete valuation, if for any $a, b \in K$, the following statements hold true: $v(ab) = v(a) + ...
13
votes
9answers
364 views

Prove that $f=x^4-4x^2+16\in\mathbb{Q}$ is irreducible

Prove that $f=x^4-4x^2+16\in\mathbb{Q}[x]$ is irreducible. I am trying to prove it with Eisenstein's criterion but without success: for p=2, it divides -4 and the constant coefficient 16, don't ...
0
votes
2answers
44 views

Tower within a Galois extension

Consider the following tower of fields: $$ K \subset M \subset L $$ If $ L/K$ is a finite Galois extension, then is it true that $ M/K $ is a Galois extension ? Is it also finite ? It is clear to ...
0
votes
1answer
19 views

Prove that the splitting field of $x^{p}-q$ for prime numbers $p,q$ is an extension of degree $p(p-1)$ in $\mathbb{Q}$.

Prove that the splitting field of $x^{p}-q$ for prime numbers $p,q$ is an extension of degree $p(p-1)$ in $\mathbb{Q}$. I know that the degree of the splitting field is bounded by $p!$, but I don't ...
1
vote
1answer
33 views

Can someone prove or help me understand the following about Euclidean fields?

Why is it that if $\delta$ and $\delta'$ both divide $\alpha$ and $\beta$, and that every $\gamma$ which divides $\alpha$ and $\beta$ also divides $\delta$ and $\delta'$, then $\delta$ and $\delta'$ ...
11
votes
3answers
409 views

Why is the collection of all algebraic extensions of F not a set?

When proving that every field has an algebraic closure, you have to be careful. In this article https://proofwiki.org/wiki/Field_has_Algebraic_Closure, and as I have been told on this site, if we have ...
2
votes
1answer
39 views

Existence of a unique subfield of degree dividing the degree of a given irreducible polynomial

Let $n$ be a positive integer and $d$ a positive integer that divides $n$. Let $\alpha \in \mathbb{R}$ be the root of the polynomial $x^{n} - 2 \in \mathbb{Q}[x]$. Prove that there is precisely one ...
1
vote
1answer
44 views

Determine the degree of the extension over Q

Determine the degree of the extension $Q(\sqrt{3+2 \sqrt{2}})$ over Q. I can see that $$3+2 \sqrt{2} = (1+ \sqrt2)(1+ \sqrt2) =(1+ \sqrt2)^2$$ does that mean $$x^2 -(1+ \sqrt2)^2)$$ has a degree $2$. ...
0
votes
1answer
11 views

Determine how many elements in $F_2[x]/(g(x))$, $F_2[x]/(h(x))$, $F_3[x]/(g(x))$ and $F_3[x]/(h(x))$

let $g(x) = x^2+x-1$ and let $h(x) = x^3-x+1$ obtain fields $4$, $8$, $9$, and $27$ elements by adjoining a root of $f(x)$ to the field $F$ where $f(x)=g(x)$ or $h(x)$ and $F = F_2$ or $F_3$. The ...