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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43 views

About a field of order $2^{n}$ with $n$ an odd integer and an additional property

I'm new in the world of fields (so I don't have any strong theorem at my disposal) and I've got stuck in this problem: Given a field of order $2^{n}$ with $n$ an odd integer and $a,b$ elements ...
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1answer
55 views

Models of ZFC as Ordered Fields

Question: Let $M$ be a set such that $\langle M,\in\rangle\vDash ZFC$. Consider the partial order $\langle M,\subseteq\rangle$. Using order extension principle there is a linear order $\subseteq^{*}$ ...
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1answer
256 views

About $\mathbb Z_{p}[\sqrt{k}]$, when is it a field? [duplicate]

I give up. I'm new in the fields world, and I'm trying to give a sufficient and necessary condition for $\mathbb{Z}_{p}[\sqrt{k}]=\{a+b\sqrt{k}:a,b\in \mathbb{Z}_{p}\}$ to be a field ($p$ is a ...
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2answers
61 views

Field extensions and gcd

Let $L|K$ be a field extension and let $u, v \in L$ be algebraic elements over $K$ such that $[K(u):K]=n$ and $[K(v):K]=m$. Show that if $\gcd(m, n)=1$ then $Irr(v, k)$ is irreducible on $K(u)$. ...
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0answers
70 views

For $a=\cos(2\pi/n)$, show that $[\mathbb{Q}(a):\mathbb{Q}] = \ldots$

For $a=\cos(2\pi/n)$, show that $[\mathbb{Q}(a):\mathbb{Q}] = \{1 \text{ for } n=1,2; (1/2)\phi(n) \text{ for } n>2\}$. $\phi$ is the Euler totient function which gives the number of coprime ...
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1answer
45 views

Does the degree of a field extension depend on the embedding of the base field?

To formulate the question more precisely, let $f$ be a field monomorphism from $F$ to $E$. The extension field $E$ can be considered a vector space over $F$, if scalar multiplication is defined by the ...
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2answers
34 views

Proof of characterization of splitting fields

I'm trying to prove that if $K$ is a finite field extension of $F$ such that $K$ is the splitting field of some collection $C$ of polynomials in $F[x]$, then every irreducible polynomial in $F[x]$ ...
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5answers
123 views

Proving a structure is a field?

Please help with what I am doing wrong here. It has been awhile since Ive been in school and need some help. The question is: Let $F$ be a field and let $G=F\times F$. Define operations of addition ...
2
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1answer
53 views

$\mathbb Q$ Field extension

Consider the Field $F = \mathbb Q(2^{\frac 1 3})$, Is $\sqrt 2 \in F$? I'm trying to figure out how to determine that and similar questions, can you give me a hint or some guidance on how to do that? ...
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1answer
45 views

understanding roots of polynomials in field extensions

I'm running into a conceptual stumbling block understanding the application of the FHT to field extensions and finding roots, if anyone has any pointers on where I might be misunderstanding. I'm ...
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1answer
69 views

Looking for a field isomorphic to $\Bbb{Q}$

I am looking for a field that is isomorphic to $\Bbb{Q}$. Could someone kindly give an example, or construct one such field?
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3answers
72 views

A question about fields containing a copy of $\Bbb{Q}$

When we say a field contains a copy of the field of rational numbers $\Bbb{Q}$, what does this really mean? Does it mean it contains a field isomorphic to $\Bbb{Q}$, or does it mean it contains ...
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3answers
234 views

Question about fields and quotients of polynomial rings

I don't see how to solve the following problem: Let $R$ be a commutative and unitary ring. If there exists a monic polynomial $f(x) \in R[x]$ so that $R[x]/(f(x))$ is a field, show that $R$ is a ...
3
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1answer
26 views

Prove that $K(\alpha)=K(\alpha^6)$ when $[K(\alpha):K]=2011$

Let $L/K$ be a finite extension and let $\alpha \in L$ so that $[K(\alpha):K]=2011$. Prove that $K(\alpha)=K(\alpha^6)$. My idea is as follows: $K \subset K(\alpha^6) \subset K(\alpha)$, therefore ...
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1answer
61 views

finite transcendence degree and algebraic closure

Let $k$ be an algebraically closed field. Let $K$ be an extension field of $k$ of finite transcendence degree over $k$. Intuitively, it seems to me that $K$ can not be algebraically closed. Is there a ...
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4answers
84 views

$\mathbb{Q}(\zeta_7)$ subextension of degree $3$

Let $\zeta_7$ be a $7$-th primitive root of unity. Is there a way to determine a subextension of $\mathbb{Q}(\zeta_7)$ that has degree $3$, without making use of Galois theory stuff?
4
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1answer
239 views

Find all subfields in extension $\mathbb{Q}\subseteq \mathbb{Q}(\sqrt[4]{2})$

I want to find all intermediate subfields of extension $\mathbb{Q}\subseteq \mathbb{Q}(\sqrt[4]{2})$. I guess that $\mathbb{Q}(\sqrt[4]{2})$ is not a splitting field, since we would have polynomial ...
6
votes
1answer
71 views

Simple field extension and roots of a polynomial

Let $K$ be a field, $f \in K[X]$ separable and irreducible with $\text{deg}(f)=n$; $x_1,...,x_n$ are the roots of $f$ in a splitting field of $f$ over $K$. Let $g \in K[X]$ be any polynomial with ...
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1answer
24 views

$\gcd(f,f')=1$ Does this imply that f has not multiply irreducible factors in $\mathbb{C}[x]$?

I want to find out if this affermation is true: let $f\in \mathbb{Q}[x]$ such that $\gcd(f,f')=1$ Does this imply that f has not multiply irreducible factors in $\mathbb{C}[x]$? (We know that it has ...
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1answer
142 views

Multiplicative Property of the degree of field extension

According to Artin's Algebra, chapter 15, section 3, the mapping property of the degree of field extension is as follows: Let $F\subset K\subset L$ be fields. Then $[L:F]=[L:K][K:F]$, where $[K:F]$ ...
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3answers
365 views

Polynomial irreducible - maximal ideal

I have a couple of ideals which I wonder if I correctly classify as maximal/prime ideal. $I_1 = \langle 2x^2 + 9x -3\rangle$, $I_2 = \langle x - 1\rangle$ $\mathbf 1)$ Is $I_1$ a maximal ideal in ...
2
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0answers
37 views

Is this proof that sigma is a fieldautomorphism legit?

I read a proof of the following theorem in "Basic abstract algebra" by Bhattacharya, Jain and Nagpaul and I thought that the proof looked overcomplicated. I have written my own proof of the theorem ...
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2answers
66 views

Let $R$ be a commutative ring with identity. Let $M$ be an ideal such that every element of $R-M$ is a unit. Then $R/M$ is a field.

Let $R$ be a commutative ring with identity. Let $M$ be an ideal such that every element of $R$ not in $M$ is a unit. Then $R/M$ is a field. I am solving this question of NBHM 2011. To solve this is ...
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1answer
162 views

Separability is transitive … for infinite extensions

Let $L/M/K$ be a tower of fields. The proof that $L/K$ is separable iff $L/M$ and $M/K$ are also separable is contained in a lot of notes and texts I've come across, subject to the assumption that the ...
4
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1answer
67 views

A Galois Extension over a field of characteristic p

Hi! I've been having a bit of trouble with this question, namely the very last part. It's from a past paper for a Galois Theory course, in which I have an exam on Monday. Basically, I can show every ...
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3answers
63 views

Basis of field - polynomial

Going through some old exams in my abstract algebra course, and was a bit curious to how I should neatly approach this problem. Let $F=\mathbb{Z_5}[x]/\langle x^3+x^2+1\rangle$ a) Give a basis of F ...
3
votes
1answer
143 views

Help with computing Galois group of $x^4 - 3$.

Let $f(x) = x^4 - 3$. I believe $Gal(f(x)) = Gal(\mathbb{Q}(\sqrt[4]{3}, i)/\mathbb{Q})$, and then we have $$\sigma_1 = \begin{cases} \sqrt[4]{3} \rightarrow \zeta^n\sqrt[4]{3}, ...
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1answer
89 views

What differences are there between $\mathbb Z_p$ and $\mathbb F_p$?

I read some books about finite fields, sometimes the author refers to the finite field $\mathbb{F}_p$ and sometimes to the finite cyclic group $\mathbb Z_p$. What is the difference between them?
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1answer
121 views

Cyclotomic Cosets and Minimal Polynomial for 45

Currently I am working on matlab in order to find Cyclotomic Cosets for 45. As 45 in not in the format of 2^m-1, matlab give me an error. I am trying to write algorithm in matlab/octave for my ...
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1answer
37 views

Is there any example of usage for a vector space over the field of formal Laurent series?

The formal Laurent series over a field is a field. Is there any example where vector spaces over that field occur naturally?
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1answer
39 views

Hilbert's Basis Theorem question

If $F$ is a field and $R = F[t_1, t_2, ... t_k]$ and $Y$ is a set of polynomials in $k$ variables over $F$ then by Hilbert's basis theorem apparently $YR = \sum\limits_{i=1}^m f_i R$ for some ...
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3answers
51 views

Prove that $\alpha_{1} ^k+ \alpha_{2} ^k +…+ \alpha_{n} ^k = n$ for $k=0$ and $0$ for $k = 1,2,…,n-1$?

For $n\geq 2$ let $\alpha_{1} + \alpha_{2} +.....+ \alpha_{n} $ be all the nth roots of unity over a field and the roots are not necessarily to be distinct. So we have to prove that $\alpha_{1} ^k+ ...
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2answers
39 views

A result on extension fields in linear algebra.

Let $F$ be a subfield of $E$, $A$ an element of $\mathcal{M}_F(m,n)$ and $b$ a vector in $F^m \subset E^m$. What is the easiest way to prove the following statement: if $Ax = b$ has a solution in ...
0
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1answer
62 views

Why is $x^2-2ux+1$, where $u = \cos(\frac{2\pi}{n})$, irreducible in $\mathbb Q(u)$?

My textbook states that $x^2-2ux+1$, where $u = \cos(\frac{2\pi}{n})$ for some $n \in \mathbb N$ is clearly irreducible in $\mathbb Q(u)$. Is this obvious? I tried to write it as a product of ...
0
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1answer
42 views

Why does the characteristic need to be 3?

and this is the solution given Why do we need the characteristic to be 3? Why wouldn't this work if over $\mathbb{Z}/\mathbb{9Z}$?
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2answers
101 views

When is a companion matrix diagonalizable and what does this say about the associated field extension?

Consider the $n\times n$ matrix $$ M=\begin{pmatrix} 0 & 0 & 0 & \cdots & 0 & 0 & -c_0\\ 1 & 0 & 0 & \cdots & 0 & 0 & -c_1\\ 0 & 1 & 0 & ...
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votes
2answers
87 views

Field $K (x)$ of rational functions over $K$, the element $x$ has no $p$th root.

Let $p$ be a prime number, and let $K = \mathbb{F}_p$. Show that in the field $K (x)$ of rational functions over $K$, the element $x$ has no $p$th root. I am having trouble understanding what $x$ ...
5
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2answers
94 views

How exactly is $i=\sqrt{-1}$ related to $\mathbb{C}$ being a closed algebraic field?

There are many known proofs of why $\mathbb{C}$ (field of complex numbers) is algebraically closed (for example Cauchy's proof ) However: how does introducing the solution to the equation $x^2 + ...
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1answer
80 views

Is there a specific method to finding a basis for vector spaces over $\mathbb{Q}$ ?

I am stuck on the first one but there are 5 questions on this so I really need help with the process. If anyone can help with any of the following. i) Find a Basis for the field K = ...
3
votes
1answer
142 views

Fixed Field of Automorphisms of $k(x)$

Fixed field of automorphisms of $k(x)$, with $k$ a field, induced by $I(x)=x$, $\varphi_1(x) = \frac{1}{1-x}$, $\varphi_2 (x)=\frac{x-1}{x}$? Since $I(x)=x$, $\varphi_1(x)=\frac{1}{1-x}$, ...
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2answers
160 views

Galois group - extend homomorphism to automorphism

Let $K \subset L$ be a finite Galois extension, $M$ a field with $K \subset M \subset L$ and $G := \text{Aut}(L/K)$. I want to show that if $\sigma \, \colon M \longrightarrow L$ is a ...
3
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1answer
89 views

Let $F/\mathbb{Q}$ be a degree 4 extension, NOT Galois. Prove that the Galois closure of $F$ has Galois group either $S_4, A_4$ or $D_8$.

The question is as the title states. So if $F=\mathbb{Q}(\alpha)$ for some alpha that satisfies a degree 4 polynomial $p(x)$, then we are looking for the splitting field of $p(x)$? I'm not sure what ...
0
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1answer
51 views

How would do this Algebra question?

$(b)$ Let $\mathbb{F}$ be a field and $X$ an indeterminate, and consider the polynomial ring $\mathbb{F}[X].$ $\quad(\mathsf{i})$ Let $f(X),g(X)\in\mathbb{F}[X]$ with $f(X),g(X)\neq0.$ Prove that ...
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1answer
25 views

Integral closure of a subring that is a polynomial ring over an algebraically closed field.

Let $K$ be an algebraically closed field that is a subring of an integral domain $D$. Assume $D$ contains an element $d$ that is transcendental over $K$. Also assume that $D$ is integral over ...
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1answer
45 views

Extending a finite field twice

Assume we have a finite field $\mathbb F_p$, an irreducible polynomial $f(x)$ of degree $m$ over $\mathbb F_p$, and an irreducible polynomial $g(y)$ of degree $n$ over $\mathbb F_p[x]/(f(x))$. Then ...
2
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2answers
108 views

What can be said about the relation between $\mathbb{Z}_p$ and $\mathbb{Z}_{(p)}$?

What can be said about the relation between $\mathbb{Z}_p$ and $\mathbb{Z}_{(p)}$? This is a question in Hungerford. I understand what both are, $\mathbb{Z}_p = \mathbb{Z}/(p)$ is a finite field and ...
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2answers
41 views

Quotient field of the intermediate integral domain

Let $R\subset T \subset F_R$, where $R,T$ are two integral domains and $F_R$ is the quotient field of $R$. I need to show that $F_T\cong F_R$. My effort: Since $T$ embeds in a field $F_R$, it must ...
3
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0answers
45 views

Is $\Bbb{Q}[\sqrt[2]{2},\sqrt[3]{2}]$ isomorphic to $\Bbb{Q}[x,y]/(x^2-y^3)$?

Say we have the field extension $\Bbb{Q}[\sqrt[2]{2},\sqrt[3]{2}]$. Is this field isomorphic to $\Bbb{Q}[x,y]/(x^2-y^3)$? I made some preliminary investigation, and this doesn't seem to be true. Is ...
0
votes
2answers
28 views

Looking at field extensions from an elementary perspective

Say an element $b$ is algebraic over $\Bbb{Q}[\sqrt{2}]$ with degree $n$. I want to prove that it is also algebraic over $\Bbb{Q}$. The proof shouldn't be along the lines of: ...
0
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2answers
76 views

Subfields of $\mathbb{Q}$

How to prove that $\mathbb{Q}$ doesn't have any proper subfields? I have no idea how to prove it.