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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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 ...
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0answers
25 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
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2answers
30 views

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

Let $K$ be a field, $G$ 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 have come to a complete ...
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0answers
18 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 ...
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22 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 ...
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1answer
58 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 ...
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0answers
51 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 $z$. ...
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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?
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2answers
36 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 ...
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1answer
17 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 ...
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0answers
24 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 $z$.
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2answers
39 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} = ...
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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 $1–3i$ ...
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1answer
19 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?
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1answer
24 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 ...
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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 $\implies f(x)=g(x)(mod\ p(x))$ ...
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1answer
21 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 ...
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2answers
34 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 $u$ ...
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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
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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 ...
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0answers
31 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 ...
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1answer
22 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 ...
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1answer
24 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 ...
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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
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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) + ...
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9answers
268 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 ...
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2answers
37 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 ...
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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 $[K:\mathbb{Q}]\leq2^{n}n!$.
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1answer
14 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 ...
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1answer
32 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 $\delta'$ ...
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3answers
384 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 ...
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0answers
24 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 ...
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1answer
41 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 $2$. ...
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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 ...
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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 ...
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1answer
46 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 ...
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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 ...
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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 $k^*$.
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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 ...
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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. ...
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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. ...
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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 ...
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129 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 ...
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1answer
21 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 ...
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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 $ ...
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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 ...
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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
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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
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1answer
34 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 ...
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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 ...