The fields tag has no wiki summary.

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### Upper bound on cardinality of a field [migrated]

Is there an upper bound on the cardinality of a field?
The "biggest" fields I know are the field of real numbers, or the field of complex numbers. Is there a field with cardinality greater than ...

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### 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.
...

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### 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
$\mathbb{C}$,
$\mathbb{R}$, ...

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302 views

### Class groups of orders

In Cox's book "Primes of the form $x^2 + ny^2$", he proves that in a quadratic imaginary field $K$, if $\mathcal O$ is an order of conductor $f \in \mathbb Z$, we have that the class group ...

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159 views

### “Degree 3 fields”

I was wondering what was known about fields $k$ having the property that any polynomial
over $k$ of degree $3$ has at least one root in $k$. Does such a field have a special name ?
Is there some kind ...

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222 views

### $L^\times / K^\times$ torsion $\Rightarrow L = K$?

Let $L/K$ be an extension of fields of characteristic zero. I want to prove that if $L^\times/K^\times$ is a torsion group (i.e. for every element $\alpha \in L$, some power of $\alpha$ lies in $K$), ...

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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 ...

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266 views

### Can non-isomorphic field extensions be isomorphic fields?

This is related to my earlier question on isomorphism of general quotients of $\:F\hspace{.02 in}[x]\:$.
Let $F$ be a field, let $p$ and $q$ be (non-zero) monic irreducible polynomials, let $I$ and ...

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### Discriminant of a compositum of number fields, a bound?

Given two number fields $E$ and $F$, is there a bound on $|d_{EF}|$, the absolute value of the absolute discriminant of the compositum of fields $EF$, in terms of $d_E$, $d_F$, and the extension ...

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82 views

### On the maximum cardinality of the image of a non-onto polynomial function on finite fields

Let $F$ be a finite field of cardinality $q$ and let $f \in F[x]$ be a non-constant polynomial of degree $d$ which is not onto (as a function from $F$ to $F$). Then how large the image of $f$ could be ...

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108 views

### Generic topology on a field

I'm wondering if there is some generic topology that can be put on any field of characteristic zero which is similar to those induced by a norm on the field. I know that for vector spaces you can take ...

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215 views

### Colorful model theory

There are a number of concepts in model theory - often situated around Hrushovski's amalgamation method (see for instance http://math.univ-lyon1.fr/~wagner/nijmegen.pdf) - which are colorfully named:
...

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207 views

### Generators of cyclic group of finite fields

Let $F$ be a finite field and $f(x)\in F[X]$ be an irreducible polynomial of degree $n$. Let $\alpha$ be a root of $f(x)$. So $E:=F[\alpha]$ is a finite field of order $|F|^{n}$.
We know that ...

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120 views

### Finite separable extension of fields imply the number of intermediate subfield is finite

The proof of statements either uses Galois theory or Artin primitive element theorem.I would like to know whether there is a proof without using these.The reason to avoid using Galois theory is that ...

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### 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?

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### 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 ...

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279 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 ...

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152 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 + ...

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### 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 ...

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478 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 ...

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### 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 ...

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### 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 ...

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### 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 ?

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### 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 ...

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262 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 ...

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### 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 ...

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269 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 ...

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### 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} ...

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### 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 + ...

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### 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 ...

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### “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 ...

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200 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 $F$?

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### 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?

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### Field extension of fields [closed]

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

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### 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 ...

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### 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 ...

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### 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

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### 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, ...

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### 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?

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### 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 ...

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### 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 $k$?

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### 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 $k(Y): k(X)$.
...

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### 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 ...

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### 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 ...

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### 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?
...

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### 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 ...

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### 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 ...

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### 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
...

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### 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.

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### “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 ...