**7**

votes

**1**answer

250 views

### Fields of characteristic zero via ultraproducts

Is every noncountable field of characteristic zero the ultraproduct (using a non principal ultrafilter over the set of prime numbers) of fields of positive characteristic?

**37**

votes

**3**answers

2k 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 ...

**3**

votes

**2**answers

357 views

### Does regular field extension preserve regularity?

Let $k$ be an arbitrary field and suppose that $K/k$ is a regular field extension. Let $V$ be regular scheme of finite type over $\text{Spec }k$ (not necessarily smooth). Is it true that $\text{Spec ...

**4**

votes

**1**answer

166 views

### Parametric septic fields $L(7) = L(3,2)$ with the same discriminant

While perusing Kluener's Database of Number Fields, I noticed that a lot of the discriminants of 7T5 came in pairs. After some doodling, I found four families. The first two are,
$$x^7 - x^6 + x^5 + ...

**0**

votes

**1**answer

166 views

### Variety with perfect function field?

My question is quite simple: Let $X$ be an irreducible algebraic variety over a field $\Bbbk$. Is there a name for such varieties with perfect function field $\Bbbk(X)$? Is this very rare? Is there ...

**4**

votes

**2**answers

499 views

### Reducing 12th degree eqns (12T179) to an 11th degree eqn

I always wondered if the fact that the quartic can be solved by a cubic can be generalized to other even degrees $n$, namely if there is an ordering of the roots $x_i$ of form ...

**14**

votes

**3**answers

953 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 ...

**29**

votes

**14**answers

4k 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 ...

**0**

votes

**1**answer

112 views

### On separable field extensions [closed]

Let $F\subseteq K$ be a finite separable field extension with $a_1,..., a_n$ an $F$-basis for $K$. Is it true that the matrix $A := [\mbox{tr}(a_ia_j)]$ is non-singular ?

**8**

votes

**3**answers

377 views

### Formulas in a Field and in a Field Extension

Let $\mathbb F$ be a field and let $a, b, c, d$ be fixed elements in the field $\mathbb F$.
Consider the formulas
1) $\exists\;x\;\;:\;\;x^2=-1.$
2) $\exists\;x\;\;:\;\;(xa=c\land xb=d).$
Formula ...

**9**

votes

**1**answer

290 views

### The j-function and Pell equations

Given the j-function,
$$j(\tau)=\frac{1}{q}+744+196884q+21493760q^2+\dots$$
it is well-known that for $\tau=\tfrac{1+\sqrt{-d}}{2}$, positive integer $d$, then $j(\tau)$ is an algebraic integer of ...

**18**

votes

**2**answers

918 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 ...

**0**

votes

**2**answers

305 views

### Rationality of curve does not depend on base change

By a curve I mean an integral one-dimensional scheme of finite type over a spectrum of a field.
Let $C$ be a curve over an arbitrary field $k$. It's probably a very well known fact, that $C$ is ...

**6**

votes

**0**answers

334 views

### Why does the Rogers-Ramanujan continued fraction $R(q)$ appear in Emma Lehmer's quintic?

Define the Ramanujan theta function $f(a,b)$ as,
$$f(a,b) = \sum_{n=-\infty}^{\infty} a^{n(n+1)/2}\,b^{n(n-1)/2}$$
and the Dedekind eta function,
$$\eta(\tau) = q^{1/24}\prod_{n=1}^{\infty} ...

**1**

vote

**0**answers

125 views

### Parametric Solvable Septics?

Known parametric solvable septics are,
$$x^7+7ax^5+14a^2x^3+7a^3x+b=0\tag{1}$$
$$x^7 - 2x^6 + x^5 - x^4 - 5x^2 - 6x - 4 + n(x - 1)x^2(x + 1)^2=0\tag{2}$$
$$x^7 + 7x^6 - 28(n^2 + 27)x^2 + 112(n^2 + ...

**16**

votes

**1**answer

783 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 ...

**4**

votes

**1**answer

194 views

### “Small” subfields of algebraically closed fields

Sufficient background:
Let $\mathcal{M}=(M,...)$ be an $\mathcal{L}$-structure and $X\subset M$.
Definition. $X$ is large if there exists a function $f:\mathcal{M}^n \overset {\leq k} \rightarrow ...

**3**

votes

**2**answers

213 views

### Subgroups of finite index of fields

Let $F$ be an infinite field and $R$ a subring of $F$. suppose that $[F:R] < \infty$ (Index of $R$ in $F$ as a subgroup is finite). Does this force $R$ to be equal to $F$?

**3**

votes

**1**answer

105 views

### Could a non-algebraically closed PAC field be a finite extension of an ordered field?

Is there such an example? Or it is known that a pseudo algebraically closed field which is a finite extension of a formally real field is algebraically closed?

**4**

votes

**1**answer

347 views

### Field extension of fields [closed]

Is the field of real numbers $\mathbb{R}$ a finite extension of some subfield $k\subset \mathbb{R}$?

**32**

votes

**1**answer

720 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 ...

**12**

votes

**2**answers

639 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 ...

**2**

votes

**1**answer

174 views

### English translation of Steinitz 1910?

Does there exist an English translation of Steinitz' 1910 work "Algebraische Theorie der Körper"?
http://www.digizeitschriften.de/dms/img/?PPN=GDZPPN002167042

**1**

vote

**0**answers

34 views

### Degree of factor of resolvent

As always with my questions this is not at research level, but the assertion is made in a research paper, plus no one's been able (or willing) to answer it over at MSE. Here is the original question, ...

**3**

votes

**2**answers

292 views

### Galois cohomology of the field of Laurent series

Let $k$ a separably closed field. Do we have that $k((t))$ is of cohomological dimension one?

**2**

votes

**0**answers

75 views

### Subfields $k\subseteq F\subseteq k(x_1,\dots,x_n)$. Is then $F\cap k[x_1,\dots,x_n]=k(f_1,\dots,f_m)\cap k[x_1,\dots,x_n]$ for polynomials $f_i$?

Let $F\subseteq k(x_1,\dots,x_n)$ be a subfield with $k\subseteq F$. I know that $F=k(\psi_1,\dots,\psi_r)$ for rational functions $\psi_i\in k(x_1,\dots,x_n)$. I'm interested in the intersection ...

**2**

votes

**0**answers

165 views

### purely non-algebraic extension that is not separable

Can you give an example of a field extension $k\subseteq K$ such that, every element of $K$ is transcendental over k and $K$ is not separable over $k$?

**3**

votes

**1**answer

120 views

### Recognizing etale covers on the level of function fields

Let $X$ be a connected, integral curve over a field $k$, and let $Y \rightarrow X$ be a finite etale cover. Corresponding to this cover there is a finite extension of function fields $k(Y): k(X)$.
...

**4**

votes

**1**answer

449 views

### Fields whose embeddings into the complex numbers are invariant under complex conjugation

Is there a general notion/description of fields $K$ such that the image of any embedding $K \hookrightarrow \mathbb{C}$ is invariant under complex conjugation, thus inducing an involution on $K$ which ...

**2**

votes

**0**answers

120 views

### Multiplicative groups in field extensions

If we let $K$ be a number field, thank to the fact that we can extend it integer ring to an UFD where the group of units is finitely generated, we can show that
$K^\ast\cong K^\ast_{tor}\times ...

**7**

votes

**0**answers

235 views

### Simple automorphism groups of field extensions of infinite transcendence degree

Let $k$ be an algebraically closed field and let $K/k$ be a field extension of infinite transcendence degree where $K$ is algebraically closed. Is it true that $\mathrm{Aut}_k(K)$ is a simple group?
...

**1**

vote

**0**answers

132 views

### Finite extensions of residue fields of Henselian DVRs

Let $K$ be an Henselian discrete valuation field such that its completion is separable over $K$. Let $F$ be its infinite residue field. Is it true that a finite extension of $F$ is a simple extension ...

**0**

votes

**1**answer

150 views

### Sums of Squares and Totally Positive Numbers

In Van der Waerden B L. Algebra Vol.I[M]. Springer, 2003, Pro. Waerden announced in page 256 that if an element $\gamma$ of a formally real field K is not a sum of squares, there exist an ordering of ...

**7**

votes

**4**answers

978 views

### The “interplay” between additive and multiplicative structure in a field

A field is an ordered triple $(F, +,\cdot)$ of a set $F$ and binary operations $+,\times$ on $F$ such that $(F,+)$ and $(F\backslash 0,\times)$ are abelian groups satisfying the distributive laws
...

**1**

vote

**2**answers

357 views

### Subfield of rational function field and which is not a rational function field

Let $K = k(x_{1}, x_{2},...,x_{n}), n\geq 2, k$ is a field. Is there exist a subfield of $K$ which is not a rational function field? Thanks.

**1**

vote

**0**answers

66 views

### “almost prime” elements in perfect Hahn field

Let $K$ be the field $\mathbb{F_p}^{alg}((\mathbb{Q}))$ (field of Hahn series over
$\mathbb{F_p}^{alg}$ and with value group $\mathbb{Q}$).
Is there elements $x$ of $K$ which are "almost prime" that ...

**2**

votes

**1**answer

199 views

### Multiple eigenvalues over imperfect fields

Let $K$ be a field. For a matrix $A\in GL_n(K)$ we can find the Jordan normal form $A'$ in $GL_n(\overline{K})$, where $\overline{K}$ is the algebraic closure of $K$. We write $j_\alpha(A)$ for the ...

**1**

vote

**0**answers

97 views

### Complementation in an extension field

If $E$ is an extension field of $F$, is $F$ necessarily (without assuming the axiom of choice) complemented as a vector subspace of $E$? (Of course the answer is easily yes if the extension is ...

**4**

votes

**1**answer

395 views

### How does an irreducible polynomial of prime power order split over an extension of prime power degree

I asked this question in a similar form on math.se here, where it has been unanswered for a little over a week some work on it can be found there. The motivation for this question was another ...

**0**

votes

**0**answers

191 views

### Pseudo-cauchy sequence and valuation

Let $k$ be a field and $x$ is transcendental over $k$. Can we construct a pseudo-cauchy sequence $(a_{i})$ convergent to $x$ with each $a_{i}$ is algebraic over $k$ and $k(a_{i})\subseteq k(a_{i + ...

**5**

votes

**1**answer

488 views

### Algorithm for determining whether two polynomials have the same splitting field

This question asks how to tell whether two cubic polynomials with coefficients in $\mathbb{Q}$ have the same splitting field. There are several answers to the question, but they don't include proofs. ...

**1**

vote

**1**answer

245 views

### Henselization of valued field

What is the importance of henselization in valuation theory, when the rank of valuation is bigger than one? Thanks

**13**

votes

**2**answers

644 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?

**9**

votes

**1**answer

454 views

### Sum of commuting semisimple operators

Let $V$ be a finite dimensional vector space over a field $K$. An operator $T:V\to V$ is called semi-simple if every $T$-invariant subspace of $V$ has a $T$-invariant complement(for algebraically ...

**0**

votes

**2**answers

251 views

### a problem about field extension [closed]

Let K and L are fields,L is a sub field of K,and L is isomorphic to K,whether can we get K=L?If true,how to prove? Thanks.

**16**

votes

**0**answers

424 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 ...

**0**

votes

**1**answer

270 views

### Algebraic closure and GIT

Does one need to work over an algebraic closed field in ordre to construct GIT quotients à la Mumford?
If yes, why?

**0**

votes

**0**answers

129 views

### Let $u \in \mathbb{Z}^{\times}_{p}$ a unit, is it true that the field $\mathbb{Q}_{p}(\zeta_{p^{n}}, \sqrt[p^{n}]{u})$ is abelian over $\mathbb{Q}_{p}$?

Let $u \in \mathbb{Z}^{\times}_{p}$ a unit, is it true that the
field $\mathbb{Q}_{p}(\zeta_{p^{n}}, \sqrt[p^{n}]{u})$ `
is abelian over $\mathbb{Q}_{p}$?

**4**

votes

**0**answers

372 views

### How arithmetical is algebraic exponentiation?

Suppose $K$ is an exponential real closed field, i.e. there is an isomorphism, say exp, between the additive group of $K$ and the multiplicative group of its positive elements.
Assume further that $Z$ ...

**3**

votes

**1**answer

212 views

### Nondegenerate involutions on the complex numbers — always just conjugation?

Suppose you have a ring homomorphism $(-)':\mathbb{C} \to \mathbb{C}$, which is an involution such that $\sum_i a_i a_i' = 0 \Leftrightarrow \forall i \ a_i=0$, where this sum is finite.
Must this ...