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4
votes
4answers
583 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 ...
0
votes
0answers
413 views

Why do we need to show that two irreducible polynomials divide each other?

Suppose we have the conjugation isomorphism $\psi_{\alpha, \beta}: F(\alpha) \mapsto F(\beta)$ defined by $$\psi_{\alpha, \beta}(a_0+a_{1} \alpha + \cdots + a_{n-1} \alpha^{n-1}) = a_0+a_{1} \beta + ...
8
votes
3answers
2k 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
315 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 ...
5
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
553 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
289 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
206 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
209 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 ...
7
votes
3answers
1k 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
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., ...
6
votes
2answers
690 views

Spheres over rational numbers and other fields

Let K be an ordered field. Define the n-sphere: $$S^n(K) := \{ (x_1,x_2,\dots,x_n+1) \in K^{n+1} \mid \sum_{i=1}^{n+1} x_i^2 = 1 \}$$ A set of vectors $v_1, v_2, \dots, v_r \in S^n(K)$ is ...
0
votes
2answers
615 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
547 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
759 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
780 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 ...
0
votes
1answer
643 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
486 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
576 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
364 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
977 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
511 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
565 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 ...
30
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 $K \setminus L$, ...
9
votes
2answers
412 views

“Kummerian” fields?

This is sort of a random, spur of the moment question, but here goes: We define [with apologies to Conan the Barbarian] a field K to be $\textbf{Kummerian}$ if there exists an index set I, and ...
14
votes
1answer
737 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 ...
8
votes
2answers
501 views

Field extension containing the eigenvectors of a Hermitian matrix

Let H be a (finite-dimensional) Hermitian matrix with algebraic numbers for its entries, all of which lie in some minimal field extension of the rational numbers; call this field ℚ(H) for short. ...
5
votes
4answers
750 views

Is the theory of incidence geometry complete?

Consider the basic axioms of planar incidence geometry, which allow us to speak of in-betweeness, collinearity and concurrency. These axioms per se are not complete, since for example, Desargues ...
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? ...
17
votes
2answers
575 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 ...
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 ...
4
votes
2answers
629 views

Notation/name for “Artin-Schreier roots”?

If x is an element of a field K and n is a positive integer, we have both a symbol and a name for a root of the polynomial t^n - x = 0: we denote it by x^{1/n} and call it an nth root of x. Of course ...
2
votes
3answers
267 views

Expressing field inclusions by polynomial equalities on coefficients

Let $A$ be the set of all quadruples $(a_0,a_1,a_2,a_3) \in {\mathbb Q}^4$ such that the polynomial $P=X^4+a_3X^3+a_2X^2+a_1X+a_0$ is irreducible and if $z$ is any root of $P$, then ${\mathbb Q}(z)$ ...
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 ...
3
votes
3answers
441 views

Is (relatively) algebraically closed stable under finite field extensions?

Let $F\subset F'$ be a field extension such that $F$ is algebraically closed inside $F'$, i.e. if $x\in F'$ is algebraic over $F$ then $x$ belongs to $F$ itself. Let now $F\subset L$ be a finite field ...
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 ...
8
votes
2answers
413 views

Evidence for Q^solv being Pseudo-algebraically-closed

This is a follow-up to the following answer: Solvable class field theory in which it is stated as a "folklore" conjecture that the maximal solvable extension of Q is pseudo algebraically closed ...
2
votes
4answers
501 views

a question on function fields (extending my previous question)

Consider the extension Q(a,b) of the field of rationals, where a,b are algebraically independent transcendentals. To Q(a,b) adjoin the roots of the polynomials x^5+a^5=1 and y^5+b^5=1. The resulting ...
10
votes
1answer
560 views

a question on function fields

Consider the transcendental extension Q(t) of the field of rationals. To Q(t) adjoin the root of the polynomial x^5+t^5=1. The resulting field Q(t)[x] is a radical extension of Q(t). Is it true that ...
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 ...
8
votes
2answers
1k views

Characterisation for separable extension of a field

Can someone verify this for me.. or tell me what reference shows me this... is this true: Let $k$ be a field. Then a field extension $K$ of $k$ is separable over $k$ iff for any field extension $L ...
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 ...
43
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 ...
-2
votes
1answer
565 views

Fundamental: Division by Zero [closed]

All the articles I've read regarding "Division by Zero" the main argument for it being an undefined operation, because all proofs lead to contradictions. ...
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 ...