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
votes
1answer
38 views

Find the cardinality of $\mathbb{F}_2$ adjoin a root of $X^4 + X + 1$

Consider the irreducible polynomial $g = X^4 + X + 1$ over $\mathbb{F}_2$ and let $E$ be the extension of $\mathbb{F}_2 =\{0,1\}$ with a root $\alpha$ of $g$. How many elements does $E$ have? ...
0
votes
3answers
88 views

Find the field of fractions and the integral closure of a subring of $\mathbb Z[x]$.

Let $R$ be a subring of $\mathbb{Z}[x]$ consisting of polynomials such that the coefficients of $x$ and $x^2$ are zero. Find the field of fractions of $R$. Find the integral closure of $R$ in it's ...
6
votes
1answer
45 views

Is there a faster way to factor $X^{12}-1$ over $\mathbb{F}_5[X]$?

Problem: Factor $X^{12}-1$ into irreducibles in $\mathbb{F}_5[X]$. This problem appeared on a past qual and took me awhile to do. While I solved it, I'll need to be able to do problems like this a ...
1
vote
0answers
45 views

How to show that $[\mathbb Q(\sqrt[3]2,\sqrt[3]5,i\sqrt 3):\mathbb Q(i\sqrt 3)]=9$

How can I show that $$[\mathbb Q(\sqrt[3]2,\sqrt[3]5,i\sqrt 3):\mathbb Q(i\sqrt 3)]=9?$$ My idea is: $\sqrt[3]2$ has for its minimal polynpmial $X^3-2$ over $\mathbb Q(i\sqrt 3)$, which I justify by: ...
5
votes
6answers
503 views

Where do the coefficients belong to?

We have the polynomial $f(x)=x^3+6x-14 \in \mathbb{Q}[x]$. We have that $f(x)$ has exactly one positive real root $a$. That means that $f(x)$ can be written as followed: $$f(x)=(x-a)(x^2+px+q)$$ ...
1
vote
3answers
64 views

annihilator polynomial of a multiplicative group in a Field?

Consider the annihilator polynomial of a multiplicative group $H$ of a field $\mathbf{F}_q$. $$A(x) = \prod_{\alpha\in H} (x-\alpha)$$ I read somewhere that this polynomial can be written as $A(x) ...
0
votes
1answer
101 views

change the matrix when we extend the field

Let $M$ be an $F_pC_q$- module represented by the matrix $$\left( \begin{matrix} a & b\\ c & d \end{matrix}\right)$$ i.e., $m_1 g=am_1 + bm_2$ and $m_2g=cm_1 + dm_2$ where g is the ...
1
vote
1answer
29 views

Two different matrix representations of complex numbers

There are two different ways to represent a complex number with $2 \times 2$ real matrices: $$ \rho: \mathbb{C} \rightarrow M_2(\mathbb{R}) \qquad \rho(z)=\rho(a+ib)= \left[ \begin{array}{ccccc} ...
2
votes
0answers
52 views

A book for advanced field theory

I am searching for an alternative text to chapter 5 of Bourbaki for field theory, that covers, for example, separable and inseparable degrees. I know the basics about field theory and Galois theory. ...
0
votes
0answers
32 views

Solution of the equation $x^r=a$

Let $F_{p^n}$ be the field with $p^n$ elements. Suppose $p^n-1=q_1^{a_1}...q_k^{a_k}$ where $q_i$ are distinct primes. Find the no. of integers $r\in\{0,1,...,p^n-2\}$ for which the equation $x^r=a$ ...
0
votes
1answer
38 views

Dual basis in a finite separable extension

I am reading the book Algebras, Rings and Modules, volume 1, by M. Hazewinkel and at the page 193 there is a proof about why the integral closure of a ring in a separable finite extension L over $k$ ...
1
vote
2answers
41 views

Show that the rings $R_1=F_5[x]/(x^2)$ and $R_2=F_5\times F_5$ are not isomorphic.

Show that the rings $R_1=F_5[x]/(x^2)$ and $R_2=F_5\times F_5$ are not isomorphic.($F_5$ is the field with $5$ elements.) My Work: Since $(0,1)$ does not have an inverse, $F_5\times F_5$ is not a ...
1
vote
1answer
33 views

Representation of Algebraic Extensions by Matrices

Let $\mathbb C$ be the field of complex numbers and $\mathbb R$ the field of real numbers. It is well known that the field $\mathbb C$ can be represented as $$\mathbb ...
13
votes
3answers
223 views

Can we construct $\Bbb C$ without first identifying $\Bbb R$?

Sometimes it is useful to consider $\Bbb C$ as our primitive and identify $\Bbb R$ as a subset of $\Bbb C$. Thus we can define $\Bbb R$ (or at least a set with all of the interesting properties of ...
0
votes
1answer
88 views

meadows and fields, aren't $0^{-1}=0$ can be proven simply from the axioms of fields?

Recall field axioms In this article http://www-compsci.swan.ac.uk/~csjvt/JVTPublications/RationalsAsADT.pdf Page 4, we have the SIP \begin{matrix} \left(-x\right)^{-1}=-\left(x^{-1}\right) \\ ...
2
votes
1answer
26 views

If $d$ divides $n$ and $\alpha$ is a root of $X^n-2$, then there is only one subfield of $\mathbb{Q}(\alpha)$ of index $d$ over $\mathbb{Q}$.

I'm studying for quals which are in a week, so I'm trying to quickly go through a ton of problems which in happier circumstances I would gladly dedicate the time to figure out myself. Let $\alpha$ be ...
1
vote
1answer
28 views

How do we conclude that $K(a,b) \subseteq \mathcal{A}_{E|K}$?

Let $K \leq E$, $\mathcal{A}_{E|K}=\{a \in E \text{ with } a \text{ algebraic } |K\}$ $K \subseteq \mathcal{A}_{E|K} \subseteq E$ We claim that $\mathcal{A}_{E|K}$ is a field. $a, b \in ...
0
votes
3answers
38 views

Solutions of $x^{p^n}=x$ form a subfield $F\subset K$.

$K$ is an algebraically closed field such that Char $K=p$. Show that the solutions of $x^{p^n}=x$ form a subfield $F\subset K$. My Work: $0$ is a solution and $0\in K$. Let $\alpha$ be a non zero ...
8
votes
2answers
168 views

How many solutions of the equation $x^2-y^2=1$ are there with $x,y\in\mathbb F_{p^n}$?

Let $p>2$ be an odd prime. Let $\mathbb F_{p^n}$ be the field with $p^n$ elements. How many solutions of the equation $x^2-y^2=1$ are there with $x,y\in\mathbb F_{p^n}$? My work: Char $F=p$. ...
3
votes
2answers
52 views

If $K \leq L$ a finite extension then it is algebraic.

I am looking at the proof of If $K \leq L$ a finite extension then it is algebraic. The proof is the following: Let $[L:K]=n<\infty$. Let $a \in L$. We will show that $\exists$ a ...
1
vote
1answer
29 views

Find the minimal irreducible polynomial $Irr(a, \mathbb{Q})$

Let the field extension $\mathbb{Q} \leq \mathbb{C}$ and $a=e^{\frac{2 \pi i}{8}}$. I have to find the minimal irreducible polynomial $Irr(a, \mathbb{Q})$. $a$ is a root of the polynomial ...
1
vote
0answers
22 views

Finding a Galois group using a cubic resolvent

I've heard there's a fast way to find the Galois group of a quartic polynomial using its resolvent. Can anyone explain how it's done or give a reference? Is this method the same as the general ...
0
votes
2answers
67 views

How to prove the polynomial $ x^{2011}-x\in \mathbb{Z}_{2011}[x] $ is separable?

I am trying to prove that the polynomial $ x^{2011}-x\in \mathbb{Z}_{2011}[x] $ is separable. Definition: Let K be a field and $f\in K[x]$ an irreducible polynomial. The polynomial $f$ is said to be ...
0
votes
2answers
41 views

Integral domain - Embedding

Let $R$ be an integral domain and the homomorphism \begin{align} \phi\colon \mathbb{Z} &\rightarrow R \\ n &\mapsto n \cdot 1_R \end{align} What does it mean that if $\ker \phi =\{0\}$ then ...
4
votes
2answers
45 views

What are the intermediate fields of $\mathbb Q(\sqrt[3]2,e^{\frac{2i\pi}{3}})$ (Galois group)

The elements of Galois group are \begin{align*} \sigma _1:\mathbb Q[\sqrt[3]2,e^{\frac{2i\pi}{3}}]&\longrightarrow \mathbb Q[\sqrt[3]2,e^{\frac{2i\pi}{3}}],\\ \sqrt[3]{2}&\longmapsto ...
4
votes
1answer
64 views

Existence of a group isomorphism between $(\mathbb K,+)$ and $(\mathbb K^\times,\cdot)$

Let $(\mathbb K,+,\cdot)$ be a field. Is there a group isomorphism between $(\mathbb K,+)$ and $(\mathbb K^\times,\cdot) $ ? The answer should clearly be negative. I tried to proceed via ...
3
votes
2answers
79 views

Adjoining a number to a field

When I studied algebra, we talked about fields such as $\mathbb{Q}[\sqrt{2}]$, the rational numbers with the square root of two adjoined to the field. Structures like these are called field extensions ...
10
votes
1answer
97 views

Subfield of $\mathbb{R}$ such that $\Bbb R/K$ is finite.

Is there a field $K \subset \mathbb{R}$ such that $1 < [\mathbb{R} : K] < \infty$? i.e a proper subfield of $\mathbb{R}$ such that the field extension $\mathbb{R}/K$ is finite.
1
vote
0answers
49 views

The number of automorphisms of a finite field

Let $M$ be a finite field and $|M| = p^s$, where $p$ is prime and $s \in \mathbb N$. Prove that the number of different isomorphisms field $M$ to $M$ equal to $s$ and this isomorphisms form a cyclic ...
4
votes
1answer
44 views

The finite field extension

Let field $K$ embedded into the finite field $M$. Prove that $M = K(\theta)$ for some $\theta \in M$. I have tried 2 ways but got stuck at both. 1) Let $|K| = p^s$ and $|M| = p^{st}$ for prime $p$ ...
1
vote
1answer
33 views

Algebraically independent equivalent conditions

I have some problems to understand the field extensions. Namely, Let $K$ be a field and $E$ its extension. Let $x_1,\ldots ,x_n$ in $E$ and $0<k<n$. Show that TFAE Family $(x_1,...,x_n)$ is ...
0
votes
3answers
91 views

Is it obvious that $\mathbb Q(\sqrt 3+\sqrt 5)=\mathbb Q(\sqrt 3,\sqrt 5)$? [duplicate]

Is it obvious that $\mathbb Q(\sqrt 3+\sqrt 5)=\mathbb Q(\sqrt 3,\sqrt 5)$ ? If not how can I show it ?
1
vote
1answer
63 views

Degree of field extension question [closed]

find the degree of the field of extension $\Bbb Q(\sqrt{2},\sqrt[4]{2},\sqrt[8]{2})$ over $\Bbb Q$. 1) 4 2) 8 3) 14 4) 32 I think it is 8.
1
vote
2answers
55 views

Finding Galois group of $x^6 - 3x^3 + 2$

I'm trying to find the Galois group of $$f(x)= x^6 - 3x^3 + 2$$ over $\mathbb{Q}$. Now I can factorise this as $$f(x) = (x-1)(x^2 + x + 1)(x^3 - 2)$$ I can see the splitting field must be ...
1
vote
1answer
24 views

Maximal algebraically independent subset and transcendence basis

I'm studying transcendence basis and I got stuck with the following problem: Let $K$ be a field and $E$ its extension. Let $S$ be a subset of $E$ such that $E$ is algebraic with respect to $K(S)$. ...
4
votes
0answers
91 views

When is $F(x+y) = F(x,y)$ for field $F$?

If $F$ is a field and $x,y$ are in an algebraic extension of $F$, I'm curious as to what we can say about $[F(x+y):F]$. I can easily prove the following:   $[F(x+y):F] \mid [F(x,y):F]$   ...
2
votes
2answers
65 views

Does there exist a subfield $S$ of $\mathbb C$ such that $\mathbb R \subset S \subset \mathbb C$?

Does there exist a subfield $S$ of $\mathbb C$ such that $\mathbb R \subset S \subset \mathbb C$ ? ; I kind of have a feeling that there does not exist any such $S$ but cannot prove . Thanks in ...
1
vote
0answers
44 views

Irreducibility of a polynomial with algebraically independent coefficients

I am learning some kind of field theory. Let $\mathbb{Q}'$ be the smallest subfield in $\mathbb{C}$ containing all roots of unity. Recently I read a book on Galois theory and met the following ...
2
votes
0answers
87 views

A question about field extension: Zariski's lemma

Suppose $E$ is a field extension of $F$ and there exists $\alpha_1,\alpha_2,\ldots,\alpha_n\in E$ such that $E=F[\alpha_1,\alpha_2,\ldots,\alpha_n]$, then the field extension $E/F$ is algebraic. Is ...
0
votes
1answer
25 views

Prove algebraic closure

Let $L/K$ be a field extension such that $L$ is algebraically closed. Show that $\{a\in L\mid[K(a):K]\lt\infty\}$ defines an algebraic closure of $K$. So this is the set of minimal polynomials ...
7
votes
2answers
108 views

Galois group of the splitting field of the polynomial $x^5 - 2$ over $\mathbb Q$

We know that the splitting field of $x^5 - 2 $ over $\mathbb Q$ is $\mathbb Q(2^{1/5}, \rho)$, where $\rho$ is a fifth root of unity. Therefore, $\left[\mathbb Q(2^{1/5} , \rho) : \mathbb Q \right] = ...
0
votes
2answers
35 views

Linear composition

can you help me with this quest? About composition $f$ and vector space $\mathbf{V}=\mathbb{Z^4_2}$ we know the following: $f \circ f = id_V$,$~~f $ $ \left(\begin{array}{ccc} 1\\ 0\\ 1\\ 0\\ ...
4
votes
2answers
51 views

When is a number square in Galois field p^n if it's not square mod p?

Here is the problem, that I'm stuck on. There is no square root of $a$ in $\mathbb{Z}_p$. Is there square root of $a$ in $GF(p^n)$? Well, it's certainly true that $$x^{p^n}=x$$ and $$x^{p^n-1}=1$$ ...
4
votes
1answer
74 views

A question on the relation of the Galois group as field automorphisms and the Galois group as permutations of roots

Let $f\in K[X]$ be a monic separable polynomial and $L$ a splitting field of $f$. Let $M=\{l_1,\ldots,l_n\}$ be the set of roots of $f$ in $L$, i.e. $$ f=(X-l_1)\cdots(X-l_n). $$ The Galois group ...
0
votes
1answer
20 views

Finiteness of a simple extension

Here I have two propositions from p.521 on Abstract Algebra written by Dummit Foote. Let $\alpha$ be algebraic over the field $F$ and let $F(\alpha)$ be the field generated by $\alpha$ over $F$. ...
0
votes
0answers
16 views

If there exists a cyclic extension of degree $p$ of $K$, then there exists a cyclic extension of degree $p^n$.

If $char K=p \neq0 $, let $K_{p}=\{ u^p-u : u\in K\}$. Show that if there exists a cyclic extension of degree $p$ of $K$, then there exists a cyclic extension of degree $p^n$ for every $n \geq 1$. ...
0
votes
1answer
27 views

Does the fixed field of automorphisms group characterize Galois extensions?

If $E/K$ is a field extension we use the notation $\def\Aut{\operatorname{Aut}}\Aut(E/K)$ for the set of field automorphisms of $E$ that are the identity over $K$. It's immediate that the set ...
1
vote
0answers
55 views

If $F$ is finite over $L_1$ and $L_2$, is it also finite over $L_1 \cap L_2$?

Let $F$ be a field and $L_1$, $L_2$ two subfields such that $F$ is finite over both $L_1$ and $L_2$. Is $F$ necessarily finite over the intersection $L_1 \cap L_2$?
1
vote
1answer
47 views

Show that $E=\mathbb{Q}(a)$

Let $f(x) \in \mathbb{Q}[x]$ an irreducible polynomial with the splitting field $E$ and let the group $Gal(E/\mathbb{Q})$ be abelian. If $a$ is a root of $f(x)$ then $E=\mathbb{Q}(a)$. Could you ...
5
votes
3answers
72 views

Is $[\bar{\mathbb Q}:\bar{\mathbb Q}\cap\mathbb R]=2$?

Is $[\bar{\mathbb Q}:\bar{\mathbb Q}\cap\mathbb R]=2$ ? I think it is true that $\bar{\mathbb Q}\cap\mathbb C=\bar{\mathbb Q}$, because I've heard that the closure of the reals is $\mathbb C$. ...