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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1answer
58 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
55 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 ...
4
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3answers
178 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 ...
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1answer
24 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 ...
1
vote
1answer
30 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
74 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
51 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
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1answer
53 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 ...
0
votes
1answer
19 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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vote
1answer
43 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
168 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 ...
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0answers
33 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
31 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
35 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
votes
1answer
46 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
41 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
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1answer
61 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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vote
1answer
72 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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vote
1answer
58 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 ...
3
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1answer
22 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
32 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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votes
3answers
47 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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votes
2answers
31 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
votes
1answer
59 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
41 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
35 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 & ...
2
votes
2answers
43 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 $x$ ...
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votes
2answers
88 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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vote
1answer
49 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 = ...
2
votes
1answer
38 views

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 $I(x)=x$, $\varphi_1(x)=\frac{1}{1-x}$, $\varphi_2 (x)=\frac{x-1}{x}$ form a group of order 3 the group is cyclic so it is generated by $\varphi_1$ then I have to find the fixed field of ...
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2answers
96 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
42 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 ...
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1answer
48 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 ...
0
votes
1answer
38 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
votes
2answers
102 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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vote
2answers
19 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 ...
2
votes
0answers
35 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
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2answers
27 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: ...
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2answers
60 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.
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0answers
21 views

Exercise 5 .39 page 46 Real and Abstract Analysis [Hewitt and Stromberg]

How to show that every non-Archimedian field contains a subfield algebraically and order isomorphic to $\mathbb{Q}(t)$, where $\mathbb{Q}(t)$ is the field of rational functions with coefficient in ...
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1answer
75 views

Is the extension Galois if $\mathrm{Aut}(K)$ acts transitively on the non-ramified prime ideals?

Let $K/\mathbb Q$ be a finite extension such that $\mathrm{Aut}(K)$ acts transitively on the prime ideals that are not ramified above the same prime $p\in\mathbb N$. Is $K$ Galois? Thanks in advance. ...
2
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0answers
66 views

Fields whose algebraic closure cannot be constructed without the axiom of choice

One can show that the statement that every field has an algebraic closure requires the axiom of choice. However, for almost all "everyday" fields, it seems that one can actually produce an algebraic ...
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1answer
32 views

Field Extensions, Subfield

Let $F$ be a subfield of a field $K$ and let $t \in K$ Let $t$ be algebraic of degree $n>1$ over $F$. Prove that $[K:F] \ge n$ Clearly there exists a polynomial $P(x)$ such that ...
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vote
3answers
66 views

Show that a map is not an automorphism in an infinite field

How should I show that a map $f(x) = x^{-1}$ for $x \neq 0$ and $f(0) = 0$ is not an automorphism for an infinite field? Thanks for any hints. Kuba
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1answer
37 views

The characteristic of the field $GF(p^n)$

How to show that characteristic of the field $GF(p^n)$ is $p$? I have come across this fact on Wikipedia webpage, but don't know how to prove it. Thanks
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1answer
32 views

complex automorphisms acting on a projective variety

Consider a complex projective variety $X=\operatorname{Proj}\frac{\mathbb C[T_1,\ldots,T_n]}{(f_1,\ldots,f_n)}$ with $f_1,\ldots,f_n$ homogeneous polynomials. If $\sigma\in\operatorname{Aut}(\mathbb ...
2
votes
1answer
39 views

Polynomials in $Z_p[x]/f(x)$

For shorthand, suppose $K=\mathbb{Z}_p[x]/f(x)$, $p$ a prime, and $\deg(f)=n$ where $f\in \mathbb{Z}_p[x]$. Then, how do we show that (1) $K$ can be written as $\mathbb{Z}_p[\theta]$, where $\theta$ ...
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1answer
83 views

Existence of an automorphism of $\Bbb C$ that fixes a finite set pointwise but does not fix $\Bbb R$ setwise

I would like to construct a (ring-theoretic) automorphism of $\Bbb C$ that fixes a finite set $A$ pointwise but does not fix $\Bbb R$ setwise. Marker's Model Theory, Corollary 1.3.6 does that in this ...
4
votes
1answer
58 views

Subgroups of $F^*$ are cyclic

Q: If $F$ a field then every finite subgroup of $F^*$ is cyclic. Solution: Suppose $d\mid |G|$ for $G$ subgroup of $F^*$ and $G$ not cyclic. Suppose $A,B$ subgroups of $G$ of order $d$. Then ...
0
votes
1answer
14 views

There are finite distinct restrictions to a subfield

Consider the field extension $L\subseteq K\subseteq \mathbb C$ where $K/L$ is finite. I must show that the set $\{\sigma_{|K}\,:\,\sigma\in\operatorname{Gal}( \mathbb C/L)\}$ is finite, but I have ...