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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0answers
33 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
26 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
24 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
16 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
21 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
37 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
11 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
17 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
23 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
23 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 $\implies f(x)=g(x)(mod\ p(x))$ ...
-1
votes
1answer
19 views

Extension over $\mathbb{Q}$ that is subfield of $n\times n$ matrices. [on hold]

Let $F$ be a field contained in the ring of $n\times n$ matrices over $\mathbb{Q}$. Prove that $[F:\mathbb{Q}]\leq n$. An earlier exercise notes that the ring of $n \times n$ matrices does contain ...
0
votes
2answers
33 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
46 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
60 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
30 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
18 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 ...
2
votes
0answers
59 views
+50

Let $R$ be a Euclidean domain with degree function $\varphi$ and $R$ not a field. If $a \ne 0$ and $b \ne 0$ be two elements in $R$…

Let $R$ be a Euclidean domain with degree function $\varphi$ and $R$ is not a field. Prove the following: (1) Let $a \ne 0$ and $b \ne 0$ be two elements in $R$. Suppose that $a\mid b$ and $b \nmid ...
1
vote
1answer
21 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
55 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
26 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) + ...
11
votes
9answers
260 views

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

Prove that $f=x^4-4x^2+16\in\mathbb{Q}$ is irreducible. I am trying to prove it with Eisenstein's criterion but without success: for p=2, it devide -4 and the constant coefficient 16, don't devide the ...
0
votes
1answer
33 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 ...
-1
votes
0answers
13 views

Degree of splitting field of palindromic polynomial [on hold]

Let $p(x)\in\mathbb{Q}[x]$ be a palindromic polynomial of even degree $2n$. Let $K$ be the splitting field of $p(x)$. Prove that $[K:\mathbb{Q}]\leq2^{n}n!$.
0
votes
1answer
13 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
28 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
380 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 ...
1
vote
0answers
23 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
37 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
10 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 ...
1
vote
1answer
15 views

Automorphism of $\mathbb{Q}({\zeta_n})/\mathbb{Q}$

I came across the theorem where, for $n=p^{a_1}\cdots p^{a_m}$: $Gal(\mathbb{Q}({\zeta_n})/\mathbb{Q})\simeq$ $Gal(\mathbb{Q}({\zeta_{p^{a_1}}})/\mathbb{Q})\times ...
1
vote
1answer
41 views

Find Primitive Root for Polynomial Field

Can someone help me get started on the problem below: Recall that $\mathbf{F}_{p^k}$ can be realized as $\mathbf{F}_p[x]/P(x) \cdot \mathbf{F}_p[x]$ where $P(x)$ is a polynomial of degree $k$ with ...
2
votes
1answer
42 views

dimension of quotient by algebraically independent elements

Let $f_1,\dots,f_s$ be algebraically independent polynomials of $A:=k[x_1,\dots,x_n]$, $s \le n$. Recall that algebraically independent means that there is no non-zero polynomial $g \in ...
0
votes
1answer
48 views

What does $k^*/k^{*^2}$ mean?

I'm trying to get a more concrete understanding of what these elements 'look like.' Here $k$ is a field, $k^*$ is multiplicative group, and $(k^*)^2$ consists of the squares in $k^*$.
1
vote
0answers
9 views

Normality of towers of fields

If I have a tower of fields $F \subseteq E \subseteq K$ and K is finite and normal over F. I know that E needn't be normal over F because for example $\mathbb{Q}(\zeta_6, \sqrt{2})$ is normal over ...
1
vote
1answer
19 views

what i want to know is how to compute the powers of $\theta$ in $F_2$ and also how many powers am i looking to compute. How can i find such powers [duplicate]

This a new chapter that we are learning and the teacher is flying through it and this are also new concept that i have just learn and i was wondering if i can have some guidance in this problem. ...
0
votes
2answers
20 views

Show that the field of real numbers has an infinite proper subfield but no finite subfields.

Show that the field of real numbers has an infinite proper subfield but no finite subfields. $\mathbb{Q}$ is an infinite subfield and as $|\mathbb{Q}| < |\mathbb{R}|$, it is also a proper subfield. ...
0
votes
1answer
15 views

Isomorphisms in characterisation of Galois extension

My definition of an extension $M/K$ to be Galois is that $Gal(M/K)$ only fixes things in K. I'm trying to prove that this is equivalent to $M/K$ being normal and separable. I know that fact that if ...
4
votes
2answers
127 views

Using Zorn's lemma to show that every field has an algebraic closure.

You may have seen that I posted this proof with some questions earlier today. But I found the answer to most of them. Now I have just one question regarding this proof, so I thought it would be better ...
0
votes
1answer
20 views

Field extension and automorphism proof understanding.

Suppose $F\subseteq L $ is any field extension, $f(x) \in F[x]$, and $b_1,b_2,...b_r$ are distinct roots of $f(x)$ in L. Prove the following statements: 1) If $\sigma$ is an automorphism of L that ...
2
votes
1answer
16 views

Infinite number of intermediate fields between K(u,v) and K

$K$ is an infinite field with char $K =p >0$ and suppose $L=K(u,v)$ where $u^p, v^p\in K $and $[L:K]=p^2$. Show that there exist infinite number of distinct intermediate fields between $K$ and $ ...
0
votes
1answer
25 views

Purely inseparable subextension of $F(x)$

Suppose that $F$ is a field of characteristic $p>0$. Prove field extension $F(x^p)\subset F(x)$ is purely inseparable. I think we should first prove that is inseparable, and then show it is ...
0
votes
0answers
35 views

By induction, show that $\sqrt{p}\notin\mathbb{Q}(\sqrt{p_1},\sqrt{p_2},\cdots,\sqrt{p_k})$ [duplicate]

By induction, show that $\sqrt{p}\notin\mathbb{Q}(\sqrt{p_1},\sqrt{p_2},\cdots,\sqrt{p_k})$ where $p_1,p_2,\cdots ,p_k,p$ are distinct primes. My try: For $k=1, ...
5
votes
1answer
37 views

$x^p -x-c$ is irreducible over a field of characteristic p if it has no root in the field

Let $c$ be an element of a field $F$ of characteristic $p$ (prime). Then how to show that $x^p -x-c$ is irreducible over $F$ if it has no root in $F$. I was trying using contradiction and by ...
4
votes
1answer
33 views

Galois Extension whose Galois Group is $\mathbb{Z}_2\oplus\mathbb{Z}_4$

The book I am using for my Abstract Algebra course is Contemporary Abstract Algebra by Joseph A. Gallian. Let $E/F$ be a Galois extension with Galois group isomorphic to ...
0
votes
2answers
29 views

“Implicit” condition about separability of a quartic polynomial

Here is an exercise in Hungerford's Algebra, page 277 Ex.12. Let $K$ be a subfield of real numbers and $f \in K[x]$ an irreducible quartic polynomial(of degree 4). If $f$ has exactly two roots, the ...
8
votes
1answer
60 views

Irreducibility of $x^{2n}+x^n+1$

I want to know for what $n$, $$x^{2n}+x^n+1$$ is irreducible modulo 2. I think for $n=3^k$ but have no idea how to prove it.
0
votes
1answer
31 views

Minimal polynomial of $\sqrt[3]{2} + \omega$ over $\mathbb{Q}.$

Is the polynomial $f(x) = x^9 - 9x^6 - 27x^3 - 27$ irreducible over $\mathbb{Q}?$ I think it is because of Eisenstein's applied to the prime $3.$ Is it the minimal polynomial of $x = e^{2 \pi i/3} + ...
1
vote
2answers
71 views

Why is F($\beta$) a subfield of F($\alpha$)?

There is a corollary in my book that says: If E is an extension field of F, $\alpha \in E$ is algebraic over $F$, and $\beta \in F(\alpha)$, then $\deg(\beta,F)$ divides $\deg(\alpha,F)$. In ...
3
votes
1answer
39 views

Find the remainder when $x^{100} + 2x + 10$ is divided by $x − 11$ in $\mathbb Z_{17}[x]$

Find the remainder when $x^{100} + 2x + 10$ is divided by $x − 11$ in $\mathbb Z_{17}[x]$ I simplified $x^{100} + 2x + 10$ to $x^{15} + 2x + 10$ and $x − 11$ to $x+6$ to be in $\mathbb Z_{17}$. ...
3
votes
0answers
31 views

Let $F,K$ be two fields $F \subset K$ and suppose $f(x),g(x) \in F[x]$ are relatively prime in $F[x].$ Prove they are relatively prime in $K[x].$

Let $F,K$ be two fields $F \subset K$ and suppose $f(x),g(x) \in F[x]$ are relatively prime in $F[x].$ Prove they are relatively prime in $K[x].$ Suppose $f(x)$ and $g(x)$ are relatively prime in ...