# Questions tagged [fields]

Fields as algebraic objects. For vector and tensor fields, use eg. [dg.differential-geometry]. For physical fields, use eg. [mp.mathematical-physics] or [quantum-field-theory].

469
questions

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### The integral closure $\overline{\mathbb{Z}}$ and the group $\overline{\mathbb{Z}}^{\times}$

Let $\mathbb{Q}$ be the field of rational numbers, and let $\overline{\mathbb{Q}}$ be its algebraic closure. Assume $\overline{\mathbb{Z}}$ is the integral closure of $\mathbb{Z}$ in $\overline{\...

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**0**answers

99 views

### Base change to algebraic closure commutes with quotient of polynomial ring by maximal ideal

Let $k$ be a field, $R:=k[x_1, \cdots , x_n]$ and $\mathfrak m$ be a maximal ideal such that $R/\mathfrak m$ is a finite separable field extension of $k$. Consider the algebraic closure $\overline k$ ...

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vote

**2**answers

294 views

### A quantity associated to a field extension

Let $F\subset E$ be a field extension. So $E$ has a natural structure of $F$-vector space.
A vector subspace $V\subset E$ is a special subspace if $F\subset V$ and $V$ is closed under the inverse ...

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votes

**0**answers

84 views

### Reduced power of an ordered field

Suppose $K$ is an ordered field, $X$ is any set, and $F$ is a filter over $X$. Let $G$ be the reduced power of $K$ by $F$. That is, we take all functions from $X$ to $K$, then take equivalence ...

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38 views

### Construction, similar to Chow's EL-numbers? Is it valid? What are the properties?

The idea of EL-numbers, proposed by Chow, impressed me very much, so I decided to build something similar and look what this will turn out.
Instead of $\exp(x)$ and $\ln(x)$ functions as the building ...

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votes

**1**answer

344 views

### Algebraic closure of $\mathbb{C}(t)$

Let $\mathbb{C}(t)$ be the field of rational functions $f(t) = \frac{p(t)}{q(t)}$ with $p,q\in\mathbb{C}[t]$.
For instance, the function $g(t) = \sqrt{t}$ does not belong to $\mathbb{C}(t)$ but is ...

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votes

**0**answers

191 views

### Metric completion of an algebraically closed field is algebraically closed?

Let $F$ be a complete metric topological field. Suppose there is a subfield $F_1 \subset F$, algebraically closed and topoolgically dense in $F$. Must $F$ itself be algebraically closed?
We can ...

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**0**answers

169 views

### Field whose absolute Galois group is $\mathbb{Z}_p$

Let $L$ be a perfect field. Assume there is an algebraic closure $L\subset \overline{L}$ and a prime $p$ such that $\mathrm{Gal}(\overline{L}/L)\cong \mathbb{Z}_p$ as topological groups.
Is there a ...

**7**

votes

**1**answer

475 views

### Is the Euler–Mascheroni constant an EL-number?

This question is based on Chow - What is a closed-form number?.
The author of the linked paper had proposed a plausible definition of "elementary numbers" (which he calls "EL-numbers&...

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votes

**1**answer

154 views

### Absolute Galois group with unique closed non-open subgroup

Is there an absolute Galois group that is not a subgroup of $\hat{\mathbb{Z}}$ and that has one and only one closed non-open subgroup?

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**1**answer

122 views

### “Violent” field isomorphisms and $p$-adic “degrees” of complex numbers

It is known that if $\mathbb{F}, \mathbb{G}$ are two algebraically closed fields with characteristic zero and equal cardinality, then $\mathbb{F}$ and $\mathbb{G}$ are isomorphic as fields. If $\...

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147 views

### Copies of the reals in $\mathbb{C}$ without the Axiom of Choice

Suppose we work in a model in which the Axiom of Choice does not hold, and in which $\mathbb{C}$ only has one nontrivial automorphism (such models exist).
Question: "how many" subfields of $\...

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**0**answers

183 views

### Are there any 1-decidable algebraic extensions of $\mathbb{Q}$ which are not decidable?

A model $M$ is decidable if the set of all first-order formulas which are true in $M$ is a recursive set. And a model is $1$-decidable if the set of all existential formulas which are true in $M$ is ...

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**0**answers

55 views

### Exponents of powers in cyclotomic extensions

For every number field $K$ and for every $a \in K$ let $$\epsilon(K;a) := \sup\!\left\{e \in \mathbb{N} : \exists b \in K \text{ such that } a = b^e\right\}. $$
Now let $K$ be a number field and let $...

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161 views

### Do algebraic completion/topological completion of fields always terminate? If so, are they unique?

Take the field $\mathbb{Q}$, If we complete it topologically with respect to the Euclidean norm, we get $\mathbb{R}$, then if we complete it algebraically, we get $\mathbb{C}$.
On the other hand, the ...

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votes

**1**answer

236 views

### What is the lowest complexity definition of $\mathbb{Z}$ in an infinite algebraic extension of $\mathbb{Q}$?

In 2009, Jochen Koenigsmann showed that $\mathbb{Z}$ is universally definable in the field $\mathbb{Q}$. And in 2012, Jennifer Park proved a result which implies that $\mathbb{Z}$ is $\exists\forall$-...

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votes

**6**answers

1k views

### Is this theory the complete theory of the real ordered field?

We know that the real ordered field can be characterized up to isomorphism as a complete ordered field. However this is a second order characterization. That raises the following question. Consider ...

**8**

votes

**1**answer

213 views

### What is the lowest complexity definition of $\mathbb{Z}$ in an infinite extension of $\mathbb{Q}$

In 2009, Jochen Koenigsmann showed that $\mathbb{Z}$ is universally definable in the field $\mathbb{Q}$. And in 2012, Jennifer Park proved a result which implies that $\mathbb{Z}$ is $\exists\forall$-...

**7**

votes

**2**answers

243 views

### Construction of Dedekind reals using higher inductive-inductive types

In the textbook Homotopy Type Theory: Univalent Foundations of Mathematics, the authors give a predicative constructive construction of the initial Cauchy complete reals $\mathbb{R}_C$ in terms of a ...

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votes

**1**answer

483 views

### What is the Galois group of one ultrapower over another ultrapower?

Let $F$ be a field, let $E$ be a field extension of $F$, and let $U$ be an ultrafilter. Then my question is, what is the relationship between the Galois groups $Gal(\Pi_U E/\Pi_U F)$ and $Gal(E/F)$?
...

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vote

**1**answer

62 views

### When is a infinite transcendence-degree rigid fields fixed by a finite extension?

A field is rigid if it has no nontrivial automorphisms. Let $F$ be a rigid field which has infinite transcendence degree over $\mathbb{Q}$, and let $E$ be a finite extension of $F$. Then my question ...

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**0**answers

145 views

### Products of short elements in a field

Consider a field $F$ of characteristic zero. Let $L=F[\alpha]$ be an extension of degree $d.$ We call an element
$$
x=x_0 + x_1 \alpha +\ldots+ x_{d-1}\alpha^{d-1}\in L
$$
short if $x_{d-1}=0.$
Under ...

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votes

**1**answer

221 views

### Hahn’s theorem on ordered fields

There is a theorem attributed to Hahn that every ordered field $F$ containing $\mathbb R$ is a subfield of a formal power series field $\mathbb R[[X^\Gamma]]$, where $\Gamma$ is an ordered abelian ...

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votes

**1**answer

246 views

### Is there a complete characterization of ordered fields without definable proper subfields?

$\mathbb{Q}$ has no proper subfields. As a result, all ordered fields elementarily equivalent to $\mathbb{Q}$ have no proper subfields which are first-order definable without parameters. And by the ...

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votes

**1**answer

261 views

### Is every field the residue field of a discretely valued field of characteristic 0?

Let $k$ be a field of positive characteristic $p$. Is there necessarily a discrete valuation ring of characteristic $0$ with maximal ideal $(p)$ and residue field isomorphic to $k$?

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votes

**1**answer

179 views

### Inverse Galois problem for non-Galois extensions

The inverse Galois problem asks whether every finite group appears as the Galois group of a Galois extension of the rational numbers.
Is anything known about the anologous problem, where the ...

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votes

**0**answers

186 views

### Are Milnor K-groups algebraic groups?

Let $k$ be a field, $K$ a finite extension of $k$, and $K_{n}^{M}(K)$ the $n$-th Milnor K-group of $K$, that is,
$$
K_{n}^{M}(K)=K^{\times}\otimes_{\mathbb{Z}}\cdots\otimes_{\mathbb{Z}} K^{\times}/I,
$...

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votes

**1**answer

309 views

### Is there a trigonometric field which is different enough from real numbers?

I found this topic in a book 'Metric Affine Geometry' by Ernst Snapper and Robert J. Troyer.
I call a field $k$ trigonometric iff there is a quadratic form $q$ over $k^2$ such that every two lines ...

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337 views

### Automorphism groups of the complex numbers, and other fields

If one accepts the Axiom of Choice (AC), then the automorphism group of $\mathbb{C}$ is a huge and wild group, very poorly understood.
But apparently if one does not accept the Axiom of Choice, then ...

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68 views

### Preservation of the Euler-Poincaré characteristics and commutativity of the multiplication in R^d

While talking about duality for polyhedra and the analogue for polygons (where the duality is essentially trivial) with a friend, I wondered if this self-duality in dimension 2 was anyhow related to ...

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64 views

### Infinite separable extensions

Let $L/K$ be an infinite algebraic separable extension of fields. One assumes that the fields are embedded in an algebraic closure $\Omega$. Consider an element $\alpha$ of $\Omega$ separable over $L$....

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57 views

### division of polynomials [closed]

Consider the set of polynomials $k[x^1,\cdots,x^n]$ over any field $k$. Now, given $p,q\in k[x^1,\cdots,x^n]$, what are the necessary and suficient conditions in order to q divide p? That is: $\exists ...

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56 views

### How to prove this integral inequality in a 2-D region?

Let $\Omega$ be a 2D region. Now we have a partial differential equation system describing the characteristics of the region:
\begin{align*} \nabla \cdot (h_0^3 P_0 \nabla P_0) &= 0 \\ \nabla \...

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102 views

### A composition of a simple extension and a separable extension is simple

Let $K/L/M$ be a tower of finite field extensions with $K/L$ separable and $L/M$ simple (in the sense of being generated by a single element). How does one show that $K/M$ is also simple?
I know that ...

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74 views

### System of linear equations in positive characteristic

Let $K$ be a field of positive characteristic $p$. Consider the system of $\mathbb F_p$-linear equations
$$\left\{\begin{array}{ccl}
a_{11}x_1+a_{12}x_2+\cdots+a_{1n}x_n&=&b_1\\
a_{11}x^p_1+a_{...

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157 views

### Issue with “definition” of pseudo algebraically closed fields

I'm having an issue with a sentence in Chapter 11 of Fried & Jarden's Field Arithmetic. As a "motto" for pseudo algebraically closed (PAC) fields, they say they are fields $K$ such that &...

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117 views

### Field of algebraic functions

We assume $K$ as a field of characteristic zero. By a field of algebraic functions of one variable over $K$ we mean a field $R$ satisfying $R=K(x,y)$ with $x$ being transcendental over $K$, and $R$ is ...

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votes

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128 views

### Isomorphism of hyperreal fields viewed as extensions of the field of reals

I asked this question on Mathematics Stackexchange but got no answer.
Question. Does $ZFC$ prove that there are non-principal ultrafilters $\mathcal U$ and $\mathcal V$ over $\mathbb N$ such that the ...

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votes

**0**answers

104 views

### Composition in function fields

Let $k=\mathbb F_q\left(\!\left(\frac1T\right)\!\right)$. One has the map: $\circ:k\times\{v\in k\mid\deg(v)>0\}\to k$ defined by $f\circ g=\sum_{n\ge-m}a_ng^{-n}$ where $f=\sum_{n\ge-m}a_n\frac1{T^...

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votes

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253 views

### Multiplicative and additive groups of the field $(\prod_{n\in\omega}\mathbb{Z}/p_n\mathbb{Z})/\simeq_{\cal U}$

Let ${\cal U}$ be a non-principal ultrafilter on $\omega$, and for each $n\in\omega$, let $p_n$ denote the $n$th prime, that is $p_0 = 2, p_1=3, \ldots$
Next we introduce the following standard ...

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150 views

### Do roots of polynomial with coefficients in a CM field lie in a CM field?

This is something that I have been thinking about for a while now, not sure if it is standard (or even true at all) or not:
Let $K/ \mathbb Q$ be a CM number field, that is, it is closed under complex ...

**6**

votes

**1**answer

203 views

### When do algebraic closures exist constructively?

The field of algebraic numbers exists constructively, since we can represent a number by an irreducible polynomial plus an estimate in rational coordinates that separates it from any other root.
More ...

**3**

votes

**1**answer

136 views

### Path integrals on statistical mechanics

In (rigorous) statistical mechanics and statistical field theory one is usually concerned in giving meaning to integrals of the form:
\begin{eqnarray}
\langle \mathcal{O}\rangle = \frac{1}{Z}\int D\...

**3**

votes

**2**answers

209 views

### Norm on tensor product of fields

Let $F$ be an algebraically closed field of characteristic $p$ equipped with an absolute value $|\cdot|:F \rightarrow \mathbb{R}_{\ge 0}$ with respect to which $F$ is complete.
Define $|\cdot|_{prod}$...

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votes

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120 views

### Central simple algebras via cohomology

I am following the book Central Simple Algebras and Galois Cohomology, by Gille and Szamuely (I am using the second edition). In section 2.4, the authors remark that the tensor product induces a ...

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votes

**1**answer

82 views

### Existence of solutions of polynomials systems (and their “rough” shape) over $\mathbb{R}$ & friends with positive-dimensional ideals

This is a follow-up (but self-contained) question to my previous one. There I asked about state-of-the-art methods to solve multivariate polynomials systems over non-algebraically closed fields in ...

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vote

**1**answer

68 views

### Under what conditions is the compositum of two rational cubic Kummer extensions a rational function field?

Let $k$ be an algebraically closed field of characteristic $p > 3$ and $F = k(x)$ be the rational function field in the variable $x$. Consider two Kummer extensions $F_1 = F(\sqrt[3]{g_1})$, $F_2 = ...

**8**

votes

**2**answers

507 views

### What is the state-of-the-art for solving polynomials systems over fields that are not algebraically closed?

I am not working in the field of algorithmic algebraic geometry - yet, for my current work, I need some results from it.
More specifically, what is the state-of-the-art when it comes to solving (...

**1**

vote

**1**answer

119 views

### Theory of extensions of non-archimedian local fields

I'm searching for a recommendable reference dealing with theory of
non-Archimedean local fields where I can find proofs of the following claims about
finite extensions $L/K$ of non-Archimedean local ...

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votes

**0**answers

59 views

### Does “tensoring” with a fixed field preserve Galois extensions of finite fields?

Let $K$ be a (possibly infinite) field of characteristic $p$, and $L$ be a finite field extension of $\mathbb{F}_p$, so that $L$ is finite and $L/\mathbb{F}_p$ is Galois. Suppose $K \otimes_{\mathbb{F}...