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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5
votes
2answers
183 views

Are $\Bbb R$ and $\Bbb C$ the only connected, locally compact fields?

I heard that $\Bbb R$ and $\Bbb C$ are the only connected, locally compact fields. Does anyone know a proof for this result?
2
votes
0answers
149 views

Galois group of CM fields

I am looking for examples of CM fields whose Galois group is not abelian. By a CM field $K$ I mean a totally imaginary quadratic extension of a totally real field $K_0$. If the extension is not Galois ...
2
votes
1answer
128 views

If $F=K(u,v)$ with $u^p$,$v^p\in K$ and $[F:K]=p^2$, $\operatorname{char} K=p>0$, then $F$ is not a simple extension of $K$.

Greetings I'm trying to show this exercise from Hungerford's Algebra Chapter five section 6 exercise 15; but I'm stuck. the exercise says the following: Let $\operatorname{char} K=p>0$ and assume ...
1
vote
1answer
61 views

Build Onto mapping of a Field to a field with an Integral Domain

The question is as follows, Let D be an integral domain. Let Q be its field of fractions and $\phi$ : D --> Q be the canonical map of D into Q. Prove that, if D is a field, then $\phi$ is ...
13
votes
2answers
335 views

A sentence false in a field of characteristic $0$ but true in all fields of positive characteristic?

Consider the language $L=\{+,\cdot, 0, 1\}$ of rings. It is easy to show using compactness that if $\sigma$ is a sentence that holds in all fields of characteristic $0$, there is some $N\in \mathbb N$ ...
2
votes
2answers
307 views

Clarifications about extension fields, its basis

I am quite confused about this topic and so I want some clarification by answering this question. Find the degree of $\mathbb{Q}$$(\sqrt[3]{2} ,\sqrt{3} )$ and its basis over the field $\mathbb{Q}$. ...
0
votes
3answers
149 views

Weird Sub-ring/field? question

Let $R=\{\frac{n}{10^{k}} \mid (n,k) \in Z, k>-1\}$ which is the sub-ring of the Rational numbers ( assumed true) Consider S a subset of R $S= \{(3/10),(33/100),(333/100),...\}$ Show that this ...
0
votes
2answers
128 views

Determine the irreducibility of a polynomial in $\mathbb{Q}[x]$

For a positive integer $n$, is the polynomial $f_n(x)=x^{n-1}+\dots+x+1$ irreducible in $\mathbb{Q}[x]$ for every $n$? irreducible for every prime $p$? irreducible for $p^n$ for all prime $p$ and ...
1
vote
1answer
113 views

irreducibilty of a polynomial over finite field

$f=x^4-x^3+14x^2+5x+16$, considering it a polynomial with coefficient in $\mathbb{F}_3$, it has no roots Considering it a polynomial with coefficient in $\mathbb{F}_3$,it is a product of two ...
5
votes
3answers
885 views

Do people ever study non-commutative fields?

I've heard of a field, and I've heard of a non-commutative (or "not-necessarily commutative) rings. Do people ever study non-commutative fields? For motivation, consider the set of all $n \times n$ ...
1
vote
3answers
254 views

Which of the following fields are isomorphic?

Let $\mathbb{Q},\mathbb{R}$ denote the fields of rational numbers and real numbers respectively. Which of the following fields are not isomorphic. (a) $\mathbb{Q}[x]/( x^2 + 1 )$ and ...
3
votes
1answer
102 views

Why is $[\overline{\mathbb{Q}}:\mathbb{Q}]$ infinite?

I hoped the question had been already asked, but I didn't find it: Why is the index of $\mathbb{Q}$ in its algebraic closure $\overline{\mathbb{Q}}$ infinite? I am aware that it can be viewed as a ...
1
vote
1answer
109 views

Unramification and compositum

The background is: a field $K$ complete with respect to a discrete valuation $|\ |$. We write $A$ and $k$ for his discret valuation ring and the residue field of $A$. We assume that $K$ and $k$ are ...
8
votes
2answers
200 views

Dimension of an algebraic closure as a vector space over its base field.

Let $k$ be an infinite field and $\bar{k}$ its algebraic closure. The Artin-Schreier Theorem tells us (among other things) that $[\bar{k}:k]=1,2,\infty$. There's a natural interpretation of ...
2
votes
0answers
133 views

Fixed points of automorphism in the field $\mathbb{C}(x,y)$

I am trying to solve a problem, and one of the parts is the following: let $M=\bigl(\begin{smallmatrix} a&b\\ c&d \end{smallmatrix} \bigr)$ be a non singular $2\times 2$ matrix with integer ...
4
votes
3answers
242 views

$E/\mathbb F_q$ extension field. Show $[\mathbb F_q(\alpha) : \mathbb F_q]$ is smallest $n$ satisfying property.

Let $q = p^m$. Suppose that $E/\mathbb F_q$ is an extension field and $\alpha \in E$ is algebraic over $\mathbb F_q$. Show that $[\mathbb F_q(\alpha) : \mathbb F_q] = $ the smallest positive integer ...
1
vote
1answer
57 views

An exponent of an element of a simple separable extension is contained in the base field.

Let $L=F(\alpha)$ be a separable extension of F, with $\alpha^k \in F$. If char$(F)=p$ and $k=p^{t}n$ and $p\not |\ n$ then $\alpha^n \in F$ Attempt at a solution: Taking the polynomial ...
5
votes
2answers
100 views

Where do I use the fact that $F$ is algebraically closed in this proof?

I have to do the following. Let $F$ be an algebraically closed field. $I\in F[X_1,...,X_n]$ an ideal. Denote by $S(I)$ the subset in $F^n$ consisting of all $n$-tuples $(a_1,...,a_n)\in F^n$ such that ...
2
votes
1answer
119 views

Splitting field for $f(x)=(x^2+x+2)(x^2+2x+2)$ over $Z_3[x]$?

Find the splitting field for $f(x)=(x^2+x+2)(x^2+2x+2)$ over $Z_3[x]$. Write $f(x)$ as a product of linear factors. This is an exercise from Gallian's Contemporary Abstract Algebra. If $Z_3[x]$ ...
5
votes
1answer
393 views

If $u$ is separable over $K$ and $v$ is purely inseparble over $K$, then $K(u,v)=K(u+v)$.

I've been having problems with this exercise of Hungerford's book, Algebra. I have been studying the section of separability by myself, and I already have a solution of the excercise, but it's very ...
3
votes
1answer
127 views

Can one construct an algebraic closure of fields like $\mathbb{F}_p(T)$ without Zorn's lemma?

I have heard that an algebraic closure of $\mathbb{Q}$ can be constructed without Zorn's lemma and so can an algebraic closure of a finite field $\mathbb{F}_p$. What about $\mathbb{F}_p(T)$? Do there ...
1
vote
0answers
49 views

Field structure of vectors in $\mathbb{R}^3$

Probably a trivial question: By representing vectors in $\mathbb{R}^2$ as complex numbers we can define multiplication of vectors so that $\mathbb{R}^2$ has a field structure. Can this be extended to ...
2
votes
2answers
146 views

Isomorphisms between $\mathbb C$ and field $\mathbb K$

There is a field $\mathbb K$. I've got an injective homomorphism $\varphi: \mathbb R \rightarrow \mathbb K$. Also I got $i \in \mathbb K$ with $i\cdot i = -1$. I have to show, that there are ...
1
vote
1answer
114 views

Ring/Field isomorphism before knowing it's a ring/field

If I know $F$ is a ring/field, and I have $G$ (unkown), but I can find a bijective map: $$\Phi:F\to G$$ such that $\forall a,b\in F.\;\Phi(a+b) =\Phi(a)+\Phi(b).\;\Phi(ab) = \Phi(a)\Phi(b)$ Does that ...
4
votes
1answer
71 views

Find separable irreducible $g$ such that $f(x)=g(x^{p^d})$

This is an exercise from VII.4. in Algebra: Chapter 0. Let $\mathcal{k}$ be a field of characteristic $p$, and $f(x)\in\mathcal{k}[x]$ an inseparable irreducible polynomial. Find a separable ...
3
votes
0answers
118 views

Question about solvability of the minimal polynomial of the element in extended field.

Let $F$ have characteristic $0$, and assume we have fields $L\supset K\supset F$. Now suppose $\alpha\in L$ be solvable by radicals over $K$, and the coefficients of the minimal polynomial of ...
0
votes
1answer
151 views

On a characterization of primitive polynomials over a finite field

Let $K$ be a finite field. Let us define a primitive polynomial as an $f \in K[X]$ s.t. the multiplicative order of $X$ in $K[X]/(f)$ is equal to $|K|^{\deg f} - 1$. I want to show that $f \in K[X]$ ...
2
votes
2answers
121 views

why $\mathbb{R}$ is a splitting field over $\mathbb{R}$

why $\mathbb{R}$ is a splitting field over $\mathbb{R}$ ? can anyone please tell me the reason for the above question though it is not a splitting field over $\mathbb{Q}$.
4
votes
3answers
525 views

Proving that $x^{2}+2$, $x^{2}-x+4$, and $x^{3}+3x-1$ are irreducible over $\mathbb{Q}$

Let $f$, $g$ and $h$ be the polynomials given by: $$f(x)=x^{2}+2$$ $$g(x)=x^{2}-x+4$$ $$h(x)=x^{3}+3x-1$$ Show that $f$, $g$ and $h$ are irreducible over $\mathbb{Q}$. I do this: ...
1
vote
1answer
226 views

splitting field of $x^p-a$ over $\mathbb{Q}$ has no primitive $p^2$ roots of unity

It is known that the splitting field of $x^p-a$ over $\mathbb{Q}$ has no $p^2$ roots of unity. We can assume $a\in \mathbb{Q}$ is not a pth power in $\mathbb{Q}$. I came up with the following proof of ...
7
votes
2answers
419 views

Compositum of fields with trivial intersection

Let $E/F$ be a finite extension. Let $L,K$ be two intermediate fields with $L\cap K = F$, and also $$[L : F] [K:F] = [E:F].$$ Must it hold that the compositum $LK$ equals $E$? If we assume that $E/F$ ...
5
votes
4answers
637 views

$F:= \{ a+b\sqrt{7} \mid a,b \in \mathbb{Q} \}$ closed under addition, subtraction, multiplication, and division

I am in my math class and I came across this problem on my past midterm. How can we prove that $F:=\{ a+b\sqrt{7} \mid a,b \in \mathbb{Q} \} $ is closed under addition, subtraction, multiplication, ...
4
votes
1answer
65 views

Polynomial Fields.

I am trying to find a polynomial in $\Bbb{Q}[x]$ which is irreducible over $\Bbb{Q}$ and has at least one linear factor over $\Bbb{R}$ and at least one irreducible quadratic factor over $\Bbb{R}$. Any ...
1
vote
2answers
267 views

Monomorphisms of a finite field extension

I have to show that if $L:K$ is a finite field extension and we have a $K$-monomorphism then this is an automorphism. I'm a bit confused by this if we have a k monomorphism: $f_K:L\rightarrow G$ for ...
5
votes
1answer
142 views

Field Extensions Problem - From Paolo Aluffi's Book

This is the exercise 4.7, chapter VII, from Paolo Aluffi's algebra book. I'm sorry for just copying the question without writing any development myself, I don't have a single ideia about how to use ...
9
votes
1answer
213 views

How to compute the Galois group of $x^5+99x-1$ over $\mathbb{Q}$?

I was stuck trying to compute the Galois group of $x^5 + 99x -1$. The problem asks to compute the Galois group over $\mathbb{F}_2, \mathbb{F}_3, \mathbb{F}_5, \mathbb{F}_{11}$ and $\mathbb{Q}$. I was ...
4
votes
4answers
126 views

Why do $f$ and $f'$ generate all of $K[X]$?

I have been studying Marcus' Number Fields. I am stuck on a remark in Appendix 2, page 258. He says: A monic irreducible polynomial $f$ of degree n over $K$ (a subfield of $\mathbb{C}$) splits into n ...
0
votes
3answers
71 views

A sum of products symmetric in the images under all the embeddings

Let $\mathbb{Q}\subset K\subset \mathbb{C}$ where $K$ is a finite extension of $\mathbb{Q}$. Let $\sigma_1, \dots, \sigma_n$ be all the embeddings $K\rightarrow \mathbb{C}$. Is it true that elements ...
2
votes
1answer
55 views

Algebraic numbers and fraction field.

If $a\in\mathbb{C}$ is algebraic, then $\mathbb{Q}(a)=\mathbb{Q}[a]$, and the converse holds too. I am having trouble proving that the same holds for several elements. Is it true that $a_1,\ldots, ...
2
votes
2answers
348 views

order of elliptic curve $y^2 = x^3 - x$ defined over $F_p$, where $p \equiv 3 \mod{4}$

It is said that the elliptic curve $y^2 = x^3 - x$ defined over a prime field $\mathbb{F}_p$, where $p \equiv 3 \mod{4}$ has an order $p + 1$. When I tried to get the elements of $E = \{(x,y) \in ...
4
votes
1answer
954 views

How to find all automorphisms of $\mathbb{Q}(\sqrt[3]{5})$? [duplicate]

Find all automorphisms of $\mathbb{Q}(\sqrt[3]{5})$. How can I solve the above problem ? Please help someone.
4
votes
1answer
278 views

Multiple roots of polynomials over a finite field

Show that $x^4+x+1$ over $\mathbb{Z}_2$ does not have any multiple zeros in any field extension of $\mathbb{Z}_2$. Show that $x^{21} + 2x^8 +1$ does not have multiple zeros in any ...
6
votes
6answers
3k views

Abstract algebra book recommendations for beginners.

I am taking abstract algebra in this semester. I find that my lecturer didn't provide enough examples for understanding. So I would like to ask what are the recommended books which has lots of solved ...
2
votes
1answer
675 views

Finding a primitive element for a field extension

Find a primitive element for the field extension $\mathbb Q(2^{1/3},2^{1/4})$ of $\mathbb Q$
4
votes
3answers
685 views

Irreducible $f(x) \in F[x]$ of prime degree, $E/F$ finite extension, $p \mid [E:F]$.

Let $F$ be a field and let $f(x) \in F[x]$ be irreducible of prime degree $p$. Let $E/F$ be a finite extension. Prove: If $f(x)$ is not irreducible in $E[x]$, then $p \mid [E:F]$. (Hint: Consider a ...
2
votes
1answer
118 views

Multiplication in $\mathbb{R}^n$

I have read that it is not possible to define multiplication in $\mathbb{R}^n$ for $n\ge 3$ in any manner whatsoever so that together with usual addition it forms a field. However I have not been able ...
3
votes
4answers
88 views

Prove that $f\sim g$ iff $f\mid g$ and $g\mid f$, where $f$ and $g$ are polynomials.

Prove that $f\sim g$ iff $f\mid g$ and $g\mid f$, where $f$ and $g$ are polynomials. It seems to me that $f$ and $g$ must be equal. And if you guys could direct me to sources that explain ...
1
vote
3answers
594 views

Two finite fields are isomorphic. [duplicate]

Let $F = \Bbb{Z}_2$. Given the irreducible polynomials $f(x)= x^3 + x + 1$, and $g(y) = y^3 + y^2 + 1$, form the fields $K = F[x]/(f(x))$ and $E = F[y] / (g(y))$. These are fields of order 8 ...
5
votes
1answer
274 views

Euclidean norms in quadratic fields

I'm currently reading a set of lecture notes of Number Theory, and there's a small part I'm having trouble understanding. A norm $N: R \rightarrow \mathbb{N} $ is Euclidean if it satisfies: for ...
0
votes
1answer
52 views

Elements in extension field

Suppose $\alpha\in\text{GF}(q^n)\setminus\text{GF}(q)$. Then there exists an irreducible polynomial $f(x)\in\text{GF}(q)[x]$ such that $f(\alpha)=0$. My question is that whether this $f(x)$ is unique ...