Fields as algebraic objects. For vector and tensor fields, use eg. [dg.differential-geometry]. For physical fields, use eg. [mp.mathematical-physics] or [quantum-field-theory].

learn more… | top users | synonyms

7
votes
2answers
475 views

An alternative description of K^*/Nm(L^*)

Is there a nice explicit description for the group $K^*/Nm_{L/K}(L^*)$ for a finite field extension $L/K$? What if for example, $L$ is obtained from $K$ by ajoining an n-th root of some $\alpha \in ...
5
votes
3answers
910 views

Is there a field which is the union of finitely many proper subfields?

Is there a field which is the union of finitely many proper subfields?
2
votes
4answers
1k views

What are examples of ordered fields that do not have the Archimedean Property?

Are the computable numbers one example?
3
votes
3answers
1k views

Some arithmetic terminology: “universal domain”, “specialization”, “Chow point”

As a non-connoisseur of arithmetic and arithmetic geometry, I would like to ask about some terminology, which meaning I haven't been able to find out on some books, nor on wikipedia, nor by google. ...
3
votes
1answer
662 views

How do you prove that a field is isomorphic to C(x)?

F is a field and F(x) is the field of rational functions in one variable x over F. Are there some clever ways to prove that some field extension K/F is (or is not) isomorphic to F(x)? One way is to ...
12
votes
2answers
884 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
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 ...
1
vote
0answers
344 views

Quadratic Solutions

There are quadratic solutions to $x^4+y^4 = z^4$ in $\mathbb{Q} (\sqrt{-7})$. But for equations such as $x^4+y^4 = nz^4$ where $n \in \mathbb{N}, \ n \neq 1$ do there still exist extension fields of ...
3
votes
1answer
726 views

How do I visualize finite covers of curves over non-algebraically closed fields?

If $L$ is algebraically closed, fields of transcendence degree one over $L$ correspond to algebraic curves over $L$ up to birational equivalence, and finite extensions correspond to finite Galois ...
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 ...
18
votes
1answer
889 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 ...
2
votes
1answer
194 views

behavior of places of a function field under automorphism

if $P_{1}$ and $P_{2}$ are distinct places of equal degree of the function field F/K, and $\sigma$ is a K-field automorphism, such that $\sigma(P_{1})=P_{2}$. then, does $\deg (P_{1}\cap K(x))=\deg ...
2
votes
3answers
403 views

Methods of showing an element is / is not in a field

Let $K$ be a field, $\alpha\in\bar{K}$, and $L/K$ a finite extension. How can we determine whether $\alpha\in L$, preferably in as much generality as possible? Of course, there may be special cases ...
5
votes
2answers
2k views

Primitive element theorem without building field extensions

Is there are nice way to prove the primitive element theorem without using field extensions? The primitive element theorem says that if $x$ and $y$ are algebraic over $F$ and $y$ is separable over ...
4
votes
0answers
137 views

Analogue of a ring extension splitting in the Kummer case

Background (the Kummer extension case) Let $R$ be a complete regular local ring (it follws that it's a UFD) with a prime integer $p$ contained in the maximal ideal of $R$ (I'm mostly interested ...
4
votes
3answers
883 views

Does there exist a Riemann surface corresponding to every field extension? Any other hypothesis needed?

My question is a doubt I had in the last point to the first answer to this MO question - "Algebraic" topologies like the Zariski topology? Can one associate a Riemann surface to any ...
1
vote
2answers
785 views

Finite extensions of field of rational functions in one variable

Let K=F(x), where x is transcendental over F and F is an algebraically closed field. Does there exist a non-commutative division algebra L with the center K and [L : K] < infinity? I think, but ...
5
votes
0answers
323 views

Point of confusion in “Topological Representations of Algebras”

Background I'm reading the article "Topological Representations of Algebras" by Arens, Kaplansky. In the proof of Theorem 6.1 we have the following situation: $X$ is a Stone space, $X_\alpha$ is a ...
3
votes
2answers
2k views

uncountable algebraically closed field other than C ?

Is there any "well-known" algebraically closed field that is uncountable other than $\mathbb{C}$ ? The algebraic closure of $\mathbb{C}(X)$ would work, but is it meaningful, is this field used in some ...
2
votes
1answer
558 views

Surjectivity of bilinear forms.

It is not uncommon to describe interesting classes of field extensions by declaring that an extension $L|K$ belongs to that class if some type of problem with $K$-coefficiens has a property over $L$ ...
125
votes
0answers
10k 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 ...
2
votes
1answer
395 views

Sum of n-th roots is rarely rational

Let $m,n$ be positive integers, and $\displaystyle \Phi_{m,n}~:~ {\mathbb{R}_+^*}^m \to \mathbb{R}_+^*, \ \ \ (x_1,x_2, \ldots , x_m) \mapsto \sum_{k=1}^m \sqrt[n]{x_k}$. Clearly for $m=1$ if for all ...
12
votes
1answer
833 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 ...
2
votes
2answers
387 views

Making the fixed field algebraically closed

K is a field and f is an automorphism of K. We're also given that the order of f, as an element of the group Aut(K), is not finite. Is it always possible to find a field L which contains K and an ...
32
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 ...
4
votes
4answers
637 views

Is a field uniquely determined by its multiplicative group/how much knows K_1 about fields?

As the title says I would like to know if $K_1(k)=k^*$ uniquely determines a field $k$. For finite fields this is clearly the case, but I suspect it is not ture in general. However I guess cooking up ...
17
votes
4answers
3k views

Is there a natural way to view the proof of Hilbert 90?

I only know of one proof of Hilbert 90, which is very smart if not magical. See for example http://hilbertthm90.wordpress.com/2008/12/11/hilberts-theorem-90the-math/ Does anyone know of a more ...
3
votes
1answer
1k views

When are intersections of finitely generated field extensions finitely generated?

Let $k$ be a field, and let $E$ and $F$ be fields extending $k$, both contained in some single extension of $k$. If $E$ and $F$ are finitely generated (as fields) over $k$, must $E\cap F$ also be ...
0
votes
2answers
320 views

roots of polynomials with multiple-of-unity coefficients implies algebraically closed?

Suppose a field has roots of all polynomials whose coefficients are 0, 0+1, 0+1+1, 0+1+1+1, 0+1+1+1+1, etc or additive inverses thereof. Is such a field necessarily algebraically closed? The ...
7
votes
3answers
1k views

For which fields K is every subring of K…?

This question was inspired by How to prove that the subrings of the rational numbers are noetherian? which some people found too routine to be of interest. So I have decided to liven things up a ...
8
votes
1answer
606 views

Existence of maximal totally ramified extensions of an arbitrary CDVF

Let $K$ be a complete, discretely valued field with (let's say) perfect residue field $k$. We have a unique maximal unramified extension $K^{unr}$ of $K$ and a unique maximal tamely ramified ...
4
votes
2answers
291 views

Slightly weakened / altered concepts of a field

I've heard of at least three slight modifications of the standard concept of field: meadow, which (according to this paper) is a commutative ring with unit equipped with a total unary operation ...
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 ...
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 ...
2
votes
1answer
214 views

Extensions of fields with lots of symmetry

[Cute question heard elsewhere] Is there a nice characterization of extensions of fields $K/k$ such that whenever $E/k$ and $E'/k$ are subextensions and $\sigma:E\to E'$ is an isomorphism over $k$, ...
1
vote
1answer
232 views

Do separable and normal have topological meanings for fields?

The terminology would suggest that a separable field extension is so because the resulting field extension has some sort of separable topology, and that a normal extension corresponds to one with a ...
8
votes
3answers
2k views

If K/k is a finite normal extension of fields, is there always an intermediate field F such that F/k is purely inseparable and K/F is separable?

I was feeling a bit rusty on my field theory, and I was reviewing out of McCarthy's excellent book, Algebraic Extensions of Fields. Out of Chapter 1, I was able to work out everything "left to the ...
16
votes
3answers
3k 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., ...
0
votes
2answers
665 views

What is the transcendence degree of Q_p and C over Q?

Is the tr.deg of Q_p over Q 1? and what about C over Q?
3
votes
1answer
670 views

Can the algebraic closure of a complete field be complete and of infinite degree?

Yes, this is yet another "foundational" question in valuation theory. Here's the background: it is a well known classical fact that the dimension (in the purely algebraic sense) of a real Banach ...
4
votes
1answer
814 views

On using field extensions to prove the impossiblity of a straightedge and compass construction

Let $z \in \mathbb{C}$. Consider the following statements: The point $z$ can be constructed with straightedge and compass starting from the points $\{ 0,1\}$. There is a field extension $K / ...
0
votes
1answer
864 views

Irreducible Polynomials in UFD and corresponding Quotient Field

Hello, "Let $D$ be a UFD and let $F$ be its quotient field. Further let $f$ be a primitive polynomial of positive degree in $D\left[x\right]$. From this it follows that that $f$ is irreducible in ...
1
vote
1answer
687 views

Orders of field automorphisms of algebraic complex numbers

Let $Aut(\bar{Q})$ be the automorphism group on the field of algebraic complex numbers. The order of an element $f \in Aut(\bar{Q})$ is the least natural number $n$ (if there exists one) such that ...
11
votes
2answers
498 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
604 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 ...
2
votes
2answers
368 views

What is it called if a vector space doesn't have an additive inverse?

so, you have, for any two members of the algebraic structure A and B and any nonnegative real values a, b: two operations: * and +, such that a*A + b*A = (a+b)*A is in the structure A + B = B + A ...
12
votes
3answers
1k 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 $\kappa$ ...
-5
votes
2answers
525 views

Field and Group Isomorphisms [closed]

We know the following isomorphism theorem for fields: Let $F, F'$ be fields. Suppose $E$ is an extension field of $F$ and $\overline{F}$ is the algebraic closure of $F$. Also $\overline{F'}$ is the ...
13
votes
2answers
613 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 ...
32
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 $K \setminus L$, ...