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 topic....

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74
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3answers
8k views

The square roots of different primes are linearly independent over the field of rationals

I need to find a way of proving that the square roots of a finite set of different primes are linearly independent over the field of rationals. I've tried to solve the problem using ...
69
votes
2answers
10k views

Example of infinite field of characteristic $p\neq 0$

Can you give me an example of infinite field of characteristic $p\neq0$? Thanks.
60
votes
6answers
5k views

Continuity of the roots of a polynomial in terms of its coefficients

It's commonly stated that the roots of a polynomial are a continuous function of the coefficients. How is this statement formalized? I would assume it's by restricting to polynomials of a fixed ...
58
votes
12answers
9k views

Do groups, rings and fields have practical applications in CS? If so, what are some?

This is ONE thing about my undergraduate studies in computer science that I haven't been able to 'link' in my real life (academic and professional). Almost everything I studied I've observed be ...
49
votes
3answers
13k views

How to find the Galois group of a polynomial?

I've been learning about Galois theory recently on my own, and I've been trying to solve tests from my university. Even though I understand all the theorems, I seem to be having some trouble with the ...
43
votes
2answers
4k views

What kind of work do modern day algebraists do?

Often times in my studies I get the impression that algebra is just a tool to help with other branches of mathematics, like algebraic geometry, algebraic number theory, algebraic topology, etc. How ...
39
votes
2answers
3k views

How to solve fifth-degree equations by elliptic functions?

From F. Klein's books, It seems that one can find the roots of a quintic equation $$z^5+az^4+bz^3+cz^2+dz+e=0$$ (where $a,b,c,d,e\in\Bbb C$) by elliptic functions. How to get that?
38
votes
2answers
587 views

Shortest irreducible polynomials over $\Bbb F_p$ of degree $n$

For any prime $p$, one can realize any finite field $\Bbb F_{p^n}$ as the quotient of the ring $\Bbb F_p[X]$ by the maximal ideal generated by an irreducible polynomial $f$ of degree $n$. By dividing ...
37
votes
6answers
3k views

In plain language, what's the significance of a field?

I just started Linear Algebra. Yesterday, I read about the ten properties of fields. As far as I can tell a field is a mathematical system that we can use to do common arithmetic. Is that correct?
35
votes
2answers
3k views

Polynomials irreducible over $\mathbb{Q}$ but reducible over $\mathbb{F}_p$ for every prime $p$

Let $f(x) \in \mathbb{Z}[x]$. If we reduce the coefficents of $f(x)$ modulo $p$, where $p$ is prime, we get a polynomial $f^*(x) \in \mathbb{F}_p[x]$. Then if $f^*(x)$ is irreducible and has the same ...
34
votes
6answers
7k views

How do I prove that $x^p-x+a$ is irreducible in a field with $p$ elements when $a\neq 0$?

How do I prove that $x^p-x+a$ is irreducible in a field with $p$ elements when $a\neq 0$? Right now I'm able to prove that it has no roots and that it is separable, but I have not a clue as to how ...
34
votes
1answer
3k views

Is there a purely algebraic proof of the Fundamental Theorem of Algebra?

Among the many techniques available at our disposal to prove FTA, is there any purely algebraic proof of the theorem? That seems reasonably unexpected, because somehow or the other we are depending ...
33
votes
6answers
7k views

What is Abstract Algebra essentially?

In the most basic sense, what is abstract algebra about? Wolfram Mathworld has the following definition: "Abstract algebra is the set of advanced topics of algebra that deal with abstract algebraic ...
33
votes
2answers
2k views

Intuitive reasoning why are quintics unsolvable

I know that quintics in general are unsolvable, whereas lower-degree equations are solvable and the formal explanation is very hard. I would like to have an intuitive reasoning of why it is so, ...
32
votes
5answers
12k views

Is $\mathbf{Q}(\sqrt{2}, \sqrt{3}) = \mathbf{Q}(\sqrt{2}+\sqrt{3})$?

Is $\mathbf{Q}(\sqrt{2}, \sqrt{3}) = \mathbf{Q}(\sqrt{2}+\sqrt{3})$ ? $$\mathbf{Q}(\sqrt{2},\sqrt{3})=\{a+b\sqrt{2}+c\sqrt{3}+d\sqrt{6} \mid a,b,c,d\in\mathbf{Q}\}$$ $$\mathbf{Q}(\sqrt{2}+\sqrt{3}...
32
votes
4answers
2k views

Fermat's Last Theorem and Kummer's Objection

In 1847 Lamé had announced that he had proven Fermat's Last Theorem. This "proof" was based on the unique factorization in $\mathbb{Z}[e^{2\pi i/p}]$. However, Kummer, proved that when $p=23$ we do ...
31
votes
2answers
527 views

Is it true in an arbitrary field that $-1$ is a sum of two squares iff it is a sum of three squares?

Here's a statement from Lam's First Course in Noncommutative Rings. (Paraphrased) Let $k$ be a field. Then the following conditions are equivalent. $$(\forall a,b,c,d\in k)\;\;(a,b,c,d)\neq 0\...
28
votes
7answers
2k views

Is $\{0\}$ a field?

Consider the set $F$ consisting of the single element $I$. Define addition and multiplication such that $I+I=I$ and $I \times I=I$ . This ring satisfies the field axioms: Closure under addition. ...
28
votes
3answers
2k views

Why isn't the inverse of the function $x\mapsto x+\sin(x)$ expressible in terms of “the functions one finds on a calculator”?

The function $f(x)=x+\sin(x)$ is easily checked to be a bijection from the reals to itself, and so it has a unique inverse $y\mapsto g(y)$ such that $f\circ g=g\circ f$ are both the identity map. ...
27
votes
2answers
1k views

Why algebraic closures?

Let me begin by summarizing the question: Why do we care about fields closed under rational exponentiation, and less about fields closed under other operations? Historically the solution for ...
26
votes
10answers
4k views

Proving $\sqrt 3$ is irrational.

There is a very simple proof by means of divisibility that $\sqrt 2$ is irrational. I have to prove that $\sqrt 3$ is irrational too, as a homework. I have done it as follows, ad absurdum: Suppose $$\...
26
votes
3answers
1k views

Does every infinite field contain a countably infinite subfield?

Does every infinite field contain a countably infinite subfield? It's easy to see that every field $K$ contains either the rational numbers $\Bbb Q$ (when $K$ has characteristic $0$) or a finite ...
26
votes
1answer
567 views

Prove that both $x+y$ and $xy$ are rational, under some conditions

As a result of the answer I got for this question - Irrational solutions to some equations in two variables - I was wondering if the next statement is always true: Let $x,y$ be real, irrational ...
26
votes
1answer
424 views

Rigidity of the category of fields

Let's call a category rigid if every self-equivalence is isomorphic to the identity. For example, $\mathsf{Set}$, $\mathsf{Grp}$, $\mathsf{Ab}$, $\mathsf{CRing}$ (MO/106838), $\mathsf{Top}$ (SE/450193)...
25
votes
4answers
1k views

What's so special about characteristic 2?

I've often read about things which do not work in a field with a characteristic $2$, mainly things which have to do with factoring, or similar things. I'm not exactly sure why, but the only example of ...
25
votes
6answers
2k views

Axiomatic characterization of the rational numbers

We have the well-known Peano axioms for the natural numbers and the real numbers can be characterized by demanding them to be a Dedekind-complete, totally ordered field (or some variation of this). ...
25
votes
2answers
887 views

Do the Liouville Numbers form a field?

The Liouville numbers are those which are better-than-polynomially approximated by rationals. More precisely, we say $x\in\mathbb{R}$ is Liouville when for all $n\in\mathbb{N}$ there is a $\tfrac pq\...
25
votes
1answer
331 views

Does there exist two non-constant polynomials $f(x),g(x)\in\mathbb Z[x]$ such that for all integers $m,n$, gcd$(f(m),g(n))=1$?

Does there exist two non-constant polynomials $f(x),g(x)\in\mathbb Z[x]$ such that for all integers $m,n$, gcd$(f(m),g(n))=1$? I think there are no such polynomials, but how to prove?
25
votes
0answers
663 views

When are nonintersecting finite degree field extensions linearly disjoint?

Let $F$ be a field, and let $K,L$ be finite degree field extensions of $F$ inside a common algebraic closure. Consider the following two properties: (i) $K$ and $L$ are linearly disjoint over $F$: ...
24
votes
6answers
1k views

Proof of Existence of Algebraic Closure: Too simple to be true?

Having read the classical proof of the existence of an Algebraic Closure (originally due to Artin), I wondered what is wrong with the following simplification (it must be wrong, otherwise why would we ...
23
votes
6answers
2k views

Is $\mathbf{C}$ the algebraic closure of any field other than $\mathbf{R}$?

It seems to me (intuitively) that there should be no other fields whose algebraic closure is $\mathbf{C}$, even though I have no reason to believe it. The facts I've been using to formulate an ...
23
votes
4answers
5k views

How to prove that the sum and product of two algebraic numbers is algebraic?

Suppose $E/F$ is a field extension and $\alpha, \beta \in E$ are algebraic over $F$. Then it is not too hard to see that when $\alpha$ is nonzero, $1/\alpha$ is also algebraic. If $a_0 + a_1\alpha + \...
23
votes
3answers
581 views

On the meaning of being algebraically closed

The definition of algebraic number is that $\alpha$ is an algebraic number if there is a nonzero polynomial $p(x)$ in $\mathbb{Q}[x]$ such that $p(\alpha)=0$. By algebraic closure, every nonconstant ...
23
votes
4answers
2k views

Algebraic closure for $\mathbb{Q}$ or $\mathbb{F}_p$ without Choice?

I know the usual proof of the existence of an algebraic closure for any field using Zorn's Lemma. The answer to this previous question makes it clear that in general, some nonconstructive axiom (not ...
23
votes
1answer
312 views

Is a field determined by its family of general linear groups?

Assume that $K,L$ are fields such that there is an isomorphism of groups $\mathrm{GL}_n(K) \cong \mathrm{GL}_n(L)$ for all $n \in \mathbb{N}$. Does it follow that $K \cong L$? I am also interested in ...
22
votes
6answers
1k views

Why should I care about fields of positive characteristic?

This is what I know about why someone might care about fields of positive characteristic: they are useful for number theory in algebraic geometry, a theory of "geometry" can be developed over them, ...
22
votes
7answers
2k views

How to prove that a complex number is not a root of unity?

$\frac35+i\frac45$ is not a root of unity though its absolute value is $1$. Suppose I don't have a calculator to calculate out its argument then how do I prove it? Is there any approach from ...
22
votes
2answers
1k views

Why is $\mathbb{Q}(t,\sqrt{t^3-t})$ not a purely transcendental extension of $\mathbb{Q}$?

This question is taken from Dummit and Foote (14.9 #6). Any help will be appreciated: Show that if $t$ is transcendental over $\mathbb{Q}$, then $\mathbb{Q}(t,\sqrt{t^3-t})$ is not a purely ...
21
votes
2answers
3k views

Wild automorphisms of the complex numbers

I read about so called "wild" automorphisms of the field of complex numbers (i.e. not the identity nor the complex conjugation). I suppose they must be rather weird and I wonder whether someone could ...
21
votes
1answer
529 views

Which fields satisfy the Freshman's Dream?

It is well-known that the Freshman's Dream, $(a+b)^p = a^p + b^p$, holds in fields of characteristic $p$. For $p = 2$, in fact those are the only fields; for, $(a+b)^2 = a^2 + b^2 \Rightarrow 2ab = 0$ ...
20
votes
2answers
753 views

What is the coproduct of fields, when it exists?

This is a slightly more advanced version of another question here. Let $\textbf{CRing}$ be the category of commutative rings with unit. Let $\textbf{Dom}$ be the category of integral domains – by ...
19
votes
3answers
16k views

what is difference between a ring and a field

The ring axioms require that addition is commutative, addition and multiplication are associative, multiplication distributes over addition. A field can be thought of as two groups with extra ...
19
votes
2answers
1k views

Are there number systems corresponding to higher cardinalities than the real numbers?

As most of you know, the set $\omega$ with cardinality $\aleph_0$ corresponds to what we normally know as the natural numbers $\mathbb{N}$, and the set $\mathcal{P}(\omega)$ with cardinality $\aleph_1$...
19
votes
3answers
2k views

How to prove that $\mathbb{Q}[\sqrt{p_1}, \sqrt{p_2}, \ldots,\sqrt{p_n} ] = \mathbb{Q}[\sqrt{p_1}+ \sqrt{p_2}+\cdots + \sqrt{p_n}]$, for $p_i$ prime?

This is Exercise 18.14 from Algebra, Isaacs. $p_{1}\ ,\ p_{2}\ ,\ ... p_{n}$ are different prime numbers. How to show that $$\mathbb{Q}[\sqrt{p_{1}}, \sqrt{p_{2}}, \ldots, \sqrt{p_{n}} ] = \mathbb{Q}[...
19
votes
1answer
431 views

When, and by whom, was “$\mathbb{C}$ is algebraically closed” dubbed the “fundamental theorem of algebra”?

Wikipedia has this enigmatic sentence on the page for the fundamental theorem of algebra: ...its name was given at a time when the study of algebra was mainly concerned with the solutions of ...
18
votes
2answers
359 views

Without the Axiom of Choice, does every infinite field contain a countably infinite subfield?

Earlier today I asked whether every infinite field contains a countably infinite subfield. That question quickly received several positive answers, but the question of whether those answers use the ...
18
votes
2answers
681 views

Atiyah's definitions of Topological Quantum Field Theory

According to Atiyah, a TQFT is a functor from the category of cobordisms to the category of vector spaces. How does this definition relate with the physics of quantum mechanics? What does the ...
18
votes
2answers
399 views

Sum of irrational numbers, a basic algebra problem

Let $x_1,\dots,x_n$ be positive rational numbers. If $\sqrt[l_1]{x_1},\dots,\sqrt[l_n]{x_n}$ are all irrational numbers (where $l_1,l_2,\dotsc,l_n\in\Bbb N^*$), does it follow that $$\sqrt[l_1]{x_1}+ ...
18
votes
2answers
704 views

field generated by a set

Let $S$ be the set of real numbers which can be written in the form $ \sum_{n\geq0}{ \frac{\epsilon_{n}}{n!}}$ ,where ${\epsilon_n}^2=\epsilon_n$ and let $K$ be the field generated by $S$ , help me ...
17
votes
3answers
2k views

Proving that $\left(\mathbb Q[\sqrt p_1,\dots,\sqrt p_n]:\mathbb Q\right)=2^n$ for distinct primes $p_i$.

I have read the following theorem. If $p_1,p_2,\dots,p_n$ are distinct prime numbers, then$$\left(\mathbb Q\left[\sqrt p_1,\dots,\sqrt p_n\right]:\mathbb Q\right)=2^n.$$ I have tried to prove a ...