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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### 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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### 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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### 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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### 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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### “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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### 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 ...

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**1**answer

223 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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### 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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### 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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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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### 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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### 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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### 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 ...

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212 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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### 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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859 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

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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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### 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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### 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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### 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 ...

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### 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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### 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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### 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 ...

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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...

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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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### 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 $Z$ ...

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### Field with cyclic product group

If a field has a cyclic multiplicative group, is it necessarily finite?

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### 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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### 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 + ...

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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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### 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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### 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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### Henselization of valued field

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

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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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### 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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### 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?