The tag has no wiki summary.

learn more… | top users | synonyms

114
votes
0answers
9k views

Ultrafilters and automorphisms of the complex field

It is well-known that it is consistent with $ZF$ that the only automorphisms of the complex field $\mathbb{C}$ are the identity map and complex conjugation. For example, we have that ...
64
votes
11answers
3k views

Can a non-surjective polynomial map from an infinite field to itself miss only finitely many points?

Is there an infinite field $k$ together with a polynomial $f \in k[x]$ such that the associated map $f \colon k \to k$ is not surjective but misses only finitely many elements in $k$ (i.e. only ...
44
votes
9answers
9k views

Galois Groups vs. Fundamental Groups

In a recent blog post Terry Tao mentions in passing that: "Class groups...are arithmetic analogues of the (abelianised) fundamental groups in topology, with Galois groups serving as the analogue of ...
34
votes
2answers
1k views

Isomorphic general linear groups implies isomorphic fields?

Suppose $n > 1$ is a natural number. Suppose $K$ and $L$ are fields such that the general linear groups of degree $n$ over them are isomorphic, i.e., $GL(n,K) \cong GL(n,L)$ as groups. Is it ...
31
votes
4answers
4k views

Fields with trivial automorphism group

Is there a nice characterization of fields whose automorphism group is trivial? Here are the facts I know. Every prime field has trivial automorphism group. Suppose L is a separable finite extension ...
31
votes
3answers
4k views

transcendental Galois theory

Suppose we define an arbitrary field extension $K/F$ to be Galois if, for all subextensions $L$ of $K/F$, we have $K^{\operatorname{Aut}(K/L)} = L$. In words: for any element $x$ of $K \setminus L$, ...
31
votes
1answer
592 views

A Topology such that the continuous functions are exactly the polynomials

(I originally asked this question on Math.SE, where it received a lot of attention, but no solution.) Which fields $K$ can be equipped with a topology such that a function $f:K \to K$ is continuous ...
27
votes
6answers
4k views

Algebraically closed fields of positive characteristic

I'm taking introductory algebraic geometry this term, so a lot of the theorems we see in class start with "Let k be an algebraically closed field." One of the things that's annoyed me is that as far ...
27
votes
3answers
1k views

Why aren't fields called “bodies” instead?

The discrepancy regarding the names of commutative division algebras in German and English has always startled me. In English they are called fields, whereas their original German name is Körper ...
27
votes
2answers
1k views

What are the possible sets of degrees of irreducible polynomials over a field?

Hopefully this is not too easy an exercise. Let $F$ be a field. Let $I \subset \mathbb{N}$ be the set of all positive integers $d$ such that there exists an irreducible polynomial of degree $d$ over ...
26
votes
2answers
908 views

If a field extension gives affine space, was it already affine space?

Let $R$ be a commutative Noetherian $F$-algebra, where $F$ is a field (perfect, say). Assume that $R \otimes_F \overline F$ is a polynomial ring over the algebraic closure $\overline F$. Does it ...
23
votes
9answers
5k views

Why are polynomials so useful in mathematics?

This is perhaps unanswerable, or perhaps I am too algebraically ignorant to phrase it cogently, but: Is there some identifiable reason that polynomials over $\mathbb{C}$, $\mathbb{R}$, ...
22
votes
5answers
1k views

Explicit elements of $K((x))((y)) \setminus K((x,y))$

In an answer to the popular question on common false beliefs in mathematics Examples of common false beliefs in mathematics. I mentioned that many people conflate the two different kinds of formal ...
20
votes
2answers
2k views

Examples of algebraic closures of finite index

So there are easy examples for algebraic closures that have index two and infinite index: $\mathbb{C}$ over $\mathbb{R}$ and the algebraic numbers over $\mathbb{Q}$. What about the other indices? ...
20
votes
4answers
2k views

What does “linearly disjoint” mean for abstract field extensions?

All definitions I've seen for the statement "$E,F$ are linearly disjoint extensions of $k$" are only meaningful when $E,F$ are given as subfields of a larger field, say $K$. I am happy with the ...
19
votes
3answers
1k views

Is the statement that every field has an algebraic closure known to be equivalent to the ultrafilter lemma?

The existence and uniqueness of algebraic closures is generally proven using Zorn's lemma. A quick Google search leads to a 1992 paper of Banaschewski, which I don't have access to, asserting that ...
19
votes
2answers
607 views

How much choice is needed to show that formally real fields can be ordered?

Background: a field is formally real if -1 is not a sum of squares of elements in that field. An ordering on a field is a linear ordering which is (in exactly the sense that you would guess if you ...
19
votes
0answers
920 views

Subfields of $\mathbb{C}$ isomorphic to $\mathbb{R}$ that have Baire property

While sitting through my complex analysis class, beginning with a very low level introduction, the teacher mentioned the obvious subfield of $\mathbb{C}$ isomorphic to $\mathbb{R}$, and I then ...
18
votes
8answers
2k views

Fantastic properties of Z/2Z

Recently I gave a lecture to master's students about some nice properties of the group with two elements $\mathbb{Z}/2\mathbb{Z}$. Typically, I wanted to present simple, natural situations where the ...
18
votes
2answers
437 views

non-isomorphic stably isomorphic fields

Q1: What is the simplest example of two non-isomorphic fields $L$ and $K$ of characteristic $0$ such that $L(x)\simeq K(x)$ (here $x$ is an indeterminate)? Q2: Do we have a sufficient criterion for ...
18
votes
1answer
860 views

What is the ring of integers of the Pythagorean field?

Following Hilbert, we call the complex numbers constructible via compass and straight-edge the field of Euclidean numbers, and the totally real such numbers the field of Pythagorean numbers. (Among ...
17
votes
3answers
1k views

Does Con(ZF) imply Con(ZF + Aut C = Z/2Z)?

How many field automorphisms does $\mathbf{C}$ have? If you assume the axiom of choice, there are tons of them -- $2^{2^{\aleph_0}}$ I believe. And what if you don't -- how essential is the axiom of ...
17
votes
7answers
5k views

Collecting proofs that finite multiplicative subgroups of fields are cyclic.

I teach elementary number theory and discrete mathematics to students who come with no abstract algebra. I have found proving the key theorem that finite multiplicative subgroups of fields are cyclic ...
17
votes
2answers
795 views

Is there an irreducible but solvable septic trinomial $x^7+ax^n+b = 0$?

The following irreducible trinomials are solvable: $$x^5-5x^2-3 = 0$$ $$x^6+3x+3 = 0$$ $$x^8-5x-5=0$$ Their Galois groups are isomorphic to ${\rm D}_5$, ${\rm S}_3 \wr {\rm C}_2$ and $({\rm S}_4 ...
16
votes
3answers
2k views

An unfamiliar (to me) form of Hensel's Lemma

In his very nice article Peter Roquette, History of valuation theory. I. (English summary) Valuation theory and its applications, Vol. I (Saskatoon, SK, 1999), 291--355, Fields Inst. Commun., ...
16
votes
1answer
672 views

On the solvable octic $x^8-x^7+29x^2+29=0$

The irreducible but solvable octic, $$x^8-x^7+29x^2+29=0\tag{1}$$ was first mentioned by Igor Schein in this 1999 sci.math post. This does not factor over a quadratic or quartic extension, but over ...
16
votes
0answers
365 views

Algebraic closure as a fibrant replacement?

Emil Artin's construction of the algebraic closure of a field $K$ is as follows. Let $K_{0} = K$, and inductively let $\{x_f\}$ be a set of indeterminates indexed by the irreducible $f$ in one ...
15
votes
12answers
2k views

Constructions unique up to non-unique isomorphism

1) Fields have algebraic closures unique up to a non-unique isomorphism. 2) Nice spaces (without base point) have universal covering spaces unique up to a non-unique isomorphism. 3) Modules have ...
15
votes
6answers
1k views

In what sense are fields an algebraic theory?

Since there is no "free field generated by a set", it would seem that 1) there is no monad on Set whose algebras are exactly the fields and 2) there is no Lawvere theory whose models in Set are ...
15
votes
4answers
2k views

FFTs over finite fields?

I'm trying to understand how to compute a fast Fourier transform over a finite field. This question arose in the analysis of some BCH codes. Consider the finite field $F$ with $2^n$ elements. It is ...
15
votes
0answers
297 views

Are the reals really a fraction field?

In an answer to this question I was led to show the trick proving that $\mathbb R$ is the fraction field of some strict subring $A\subsetneq \mathbb R=\operatorname{Frac}(A)$. A crucial point in the ...
14
votes
1answer
743 views

When f(x)-a and f(x)-b yield the same field extension

An interesting mathoverflow question was one due to Philipp Lampe that asked whether a non-surjective polynomial function on an infinite field can miss only finitely many values. In my interpretation ...
14
votes
3answers
698 views

Which fields have multiplicative group isomorphic to additive group times Z/2Z?

Let $K$ be a field, $K_{*}$ its multiplicative group and $K_{+}$ its additive group. As Richard Stanley notes in this answer, the only field for which $K_{+} \simeq K_{*} \times ...
14
votes
1answer
551 views

Galois theory: Generalization of Abel’s Theorem? (Better version!)

(Unintentionally I have previously asked a similar and perhaps in itself not uninteresting question Galois theory: Generalization of Abel's Theorem? but this is what I originally had in mind.) ...
14
votes
0answers
651 views

Is there an infinite field of characteristic 2 whose multiplicative group is torsion free and (direct-sum) indecomposable?

Let $F$ be a infinite field of characteristic 2 whose multiplicative group $F^*$ is torsion free. I would like to conclude that $F^*$ is decomposable or find an example where $F^*$ is indecomposable. ...
13
votes
2answers
602 views

The Surreal numbers satisfy all the field axioms except that its elements constitute a proper class. Is it safe to call it a field?

Do all the field theorems apply to surreal numbers? If fields were redefined so that their elements were allowed to come from an arbitrary class, would the theory look different to an algebraist?
13
votes
1answer
643 views

Is every algebraic extension of a field of absolute transcendence degree one a separable extension of a purely inseparable extension?

Any decent course on field theory will state that in characteristic $p$ an extension of fields $k\subset K$ canonically decomposes as the tower $k\subset K_{sep}\subset K$ with $K$ purely inseparable ...
13
votes
2answers
597 views

What is the prime spectrum of a Cauchy series ring?

Let $k$ be a field, and let $| \ |$ be a norm on $k$. The norm induces a metric. To construct the completion $\hat{k}$ as a normed field, the standard recipe is to take the quotient of the ring ...
12
votes
2answers
859 views

Is completeness of a field an algebraic property?

Pretty straitforward: If a field has a metric in which it is complete can it have a metric in which it is not complete? By metric I mean field norm of course
12
votes
3answers
990 views

Are there as many real-closed fields of a given cardinality as I think there are?

Let $\kappa$ be an infinite cardinal. Then there exists at least one real-closed field of cardinality $\kappa$ (e.g. Lowenheim-Skolem; or, start with a function field over $\mathbb{Q}$ in $\kappa$ ...
11
votes
3answers
4k views

Finite extension of fields with no primitive element

What is an example of a finite field extension which is not generated by a single element? Background: A finite field extension E of F is generated by a primitive element if and only if there are a ...
11
votes
2answers
1k views

Are the field norm and trace the unique “nice” maps between fields?

This seems like an obvious fact, but I'm not sure what the necessary meaning of "nice" is to get a result like this. I'm wondering if there is a theorem of the form: For any <1> field extension ...
11
votes
3answers
1k views

Intuition for Model Theoretic Proof of the Nullstellensatz

I recently read the model-theoretic proof of the Nullstellensatz using quantifier elimination (see www.msri.org/publications/books/Book39/files/marker.pdf). I'm convinced that the Nullstellensatz is ...
11
votes
2answers
472 views

To what extent can fields be classified?

The study of algebraic geometry usually begins with the choice of a base field $k$. In practice, this is usually one of the prime fields $\mathbb{Q}$ or $\mathbb{F}_p$, or topological completions and ...
11
votes
1answer
805 views

Is -1 a sum of 2 squares in a certain field K?

Consider the field of fractions $K$ of the quotient algebra $\mathbb{R}[x,y,z,t]/(x^2+y^2+z^2+t^2+1)$, where $\mathbb{R}$ is the field of real numbers and $x,y,z,t$ are variables. Clearly $-1$ is a ...
11
votes
2answers
487 views

Algebraicity of the completion of a field? Finiteness?

At the end of my 8410 class today (see http://www.math.uga.edu/~pete/MATH8410.html if you care), one of my students asked me the following very interesting question: Let $(K,|\ |)$ be a normed field, ...
11
votes
2answers
582 views

Which commutative groups are the group of units of some field?

Inspired by a recent question on the multiplicative group of fields. Necessary conditions include that there are at most $n$ solutions to $x^n = 1$ in such a group and that any finite subgroup is ...
11
votes
1answer
463 views

torsion group of the multiplicative group of a field

Let $F$ be any field of zero characteristic, $F^{\ast}$ its multiplicative group and $T$ is the torsion group. Is it true that $T$ is a direct summand for $F^{\ast}$?
11
votes
2answers
1k views

Automorphisms of $\mathbb{C}$

Is it true that $G_{\mathbb{Q}}$, the absolute Galois group of $\mathbb{Q}$, is a subgroup of $Aut(\mathbb{C})$ ? Or a simpler question: can any automorphism of $\overline{\mathbb{Q}}$ be extended to ...
11
votes
1answer
580 views

Converse to Banach’s fixed point theorem for ordered fields?

Suppose $R$ is an ordered field. Call a continuous map $f: R \rightarrow R$ a contraction if there exists $r < 1$ (in $R$) such that $|f(x)-f(y)| \leq r |x-y|$ for all $x,y \in R$ (where $|x| := ...