**2**

votes

**1**answer

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

**1**answer

240 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

**3**answers

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 ...

**17**

votes

**3**answers

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

**2**answers

701 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?

**4**

votes

**1**answer

734 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

**1**answer

843 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

**1**answer

908 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

**1**answer

703 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

**2**answers

507 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, ...

**12**

votes

**2**answers

820 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

**2**answers

377 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

**3**answers

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 $\kappa$ ...

**6**

votes

**1**answer

444 views

### Split powers of the multiplicative group of a field

Let $K$ be a field, $K^\times$ its multiplicative group and $I$ an infinite set. Is then $(K^\times)^{(I)} \subseteq (K^\times)^I$ a direct summand? If not, is it possible to characterize the fields ...

**-5**

votes

**2**answers

532 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

**2**answers

632 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

**3**answers

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 $K \setminus L$, ...

**10**

votes

**2**answers

447 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 ...

**15**

votes

**1**answer

765 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

**2**answers

526 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

**4**answers

815 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 ...

**21**

votes

**2**answers

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?
...

**22**

votes

**2**answers

725 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 ...

**24**

votes

**4**answers

3k 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

**2**answers

678 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

**3**answers

270 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

**2**answers

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

**3**answers

466 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 ...

**66**

votes

**11**answers

4k 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

**2**answers

426 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

**4**answers

505 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

**1**answer

572 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 ...

**29**

votes

**6**answers

5k 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

**2**answers

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 ...

**16**

votes

**6**answers

2k 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 ...

**44**

votes

**9**answers

11k 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 ...

**-3**

votes

**1**answer

625 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.
...

**16**

votes

**4**answers

5k 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 ...