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].

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How hard must "no high-degree irreducibles" proofs be?

Let $\mathsf{RCF}$ be the usual theory of real closed fields and for $n>2$ let $\theta_n$ be the sentence "No degree-$n$ polynomial is irreducible." Since $\mathsf{RCF}$ is complete, for ...
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7 votes
1 answer
330 views

Field extensions over which algebraic varieties cannot acquire points

The following fact (slightly reworded here) is proven in this answer: If $K$ is a purely transcendental extension of an infinite¹ field $k$, then whenever a separated scheme $X$ of finite type over $...
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1 vote
1 answer
135 views

A "semi-genetic" definition of addition and multiplication in the field $\operatorname{On}_p$?

Let $+,\cdot$ denote multiplication in $\mathbb{N}_0$. The addition and multiplication in $\operatorname{On}_p$ are denoted $\oplus, \otimes$. Recursive definition of addition: $$x \oplus y := ((x+y) \...
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2 votes
0 answers
58 views

What's the number of irreducible polynomials over rational numbers with constraint on coefficients?

Suppose you have polynomials $f(x) = \sum\limits_{k=0}^n a_k x^k$ where $f \in \mathbb{Q}[x]$ and $a_k$ can be either $0$ or $1$. There are $2^n$ such polynomials in total. How can I find a number of ...
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7 votes
1 answer
369 views

Factorization of an irreducible polynomial in the field extension it defines

In field theory, the following fact is used in the construction of splitting fields: Given a field $F$ and an irreducible polynomial $f \in F[x]$, the quotient $F[\alpha]/(f(\alpha))$ is a field ...
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3 votes
1 answer
103 views

How to determine the degree of a rational function field over a relatively algebraic subfield?

Let $K$ be a field and $K(x_1,\cdots,x_n)$ be the degree-$n$ purely transcendental extension of $K$. Given homogeneous polynomials $f_1,\cdots,f_n\in K[x_1,\cdots,x_n]\setminus K$ with $\deg f_i=d_i$,...
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  • 301
12 votes
0 answers
327 views

Why is it so hard to give examples of differentially closed fields?

The theory of algebraically closed field, say in characteristic zero, and of differentially closed fields (of characteristic zero) have much in common: quantifier elimination and (hence) decidability; ...
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3 votes
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96 views

What direction does the derivation of an inseparable algebraic variable point in?

I've been thinking about the geometry of inseparable field extensions lately, since I'm studying smoothness in commutative rings in an advanced topics course this semester. I've generally come to the ...
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1 vote
2 answers
171 views

In the Levi-Civita field, are there elements such that the standard parts of their subsequent powers produce an arbitrary sequence?

In the Levi-Civita field, are there elements such that the standard parts of their subsequent powers produce an arbitrary sequence? Particularly, is there an element $w$ of the field such that the ...
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1 vote
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61 views

Indeterminacy locus of an algebraic function

Let $K=\mathbb{C}(t_1,\dots,t_n)$ be the field of rational functions, $f$ an algebraic function over $K$ and assume the field extension $K(f)/K$ is non-solvable. Is it possible to characterise the ...
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4 votes
1 answer
242 views

Common Galois extension over $\mathbb Q $

Suppose $L'$ is a fixed cyclic galois extension over $\mathbb {Q} $ of degree $4$.Now we know that there exists also a degree $k$ extension $L$ over $L'$ but the extension $(L/\mathbb{Q})$ may not ...
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1 vote
2 answers
261 views

A variation on Abhyankar–Moh–Suzuki theorem

The well-known theorem of Abhyankar–Moh–Suzuki says the following: Let $f=f(t), g=g(t) \in k[t]$, $k$ is a field of characteristic zero. If $k[f,g]=k[t]$, then $\deg(f) \mid \deg(g)$ or $\deg(g) \mid \...
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  • 2,573
10 votes
1 answer
290 views

Proving that polynomials belonging to a certain family are reducible

In an article, I've found the following result. Unfortunately, it was derived from a general, somewhat complicated theory, that would be cumbersome for this result alone. Assume that $\mathbb F_p$ is ...
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  • 567
5 votes
1 answer
186 views

Lines on quadric surfaces

Consider a smooth quadric surface $Q\subset\mathbb{P}^3$ over a field $k$. Are there natural hypotheses one can put on $k$ in order to ensure the existence of a line defined over $k$ on $Q$?
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  • 387
2 votes
0 answers
107 views

Abelian approximation of fields

Given a field extension $K/k$ of finite degree and a norm $d$ on $\overline{k}$, what is the smallest real number $\alpha_{d}^K$ such that for every element $z$ of $K$ there is an element $z^a$ of $k^...
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  • 21
3 votes
1 answer
139 views

A question about decompositions of rational functions

Let $f_1,g_1,f_2,g_2$ be non-constant rational functions on the Riemann sphere (i.e. elements of $\Bbb{C}(z)-\Bbb{C}$) satisfying $f_1\circ g_1=f_2\circ g_2$. Suppose there is a prime number $p$ such ...
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31 votes
3 answers
2k views

Is the hierarchy of relative geometric constructibility by straightedge and compass a dense order?

Consider the hierarchy of relative geometric constructibility by straightedge and compass. Namely, given a geometric figure $B$, a set of points in the plane, we define that geometric figure $A$ is ...
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11 votes
2 answers
842 views

Algebraic topology over fields other than ${\bf R}$

Is there an algebraic topology for spaces defined on fields other than ${\bf R}$, including totally discontinuous fields ? By this, I am not talking about the field of coefficients, but about the ...
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1 vote
2 answers
172 views

If $G=\mathsf{Aut}_k(F)$ acts on field $F$ algebraic over $k$ then do we have: orbit $G\alpha=\text{ roots of minimal polynomial of }\alpha$?

I posed this question on Math.Stackexchange (see here) but until now there was no response. This made me decide to give it a try here. Let $k\subseteq F$ denote an algebraic field extension and let $\...
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4 votes
0 answers
198 views

Topological field isomorphic to $\mathbb{C}$

Let $K$ be a topological field. If $K$ is a connected Hausdorff space, and is algebraically closed, is it true that $K$ is isomorphic to $\mathbb{C}$ ? (I have deleted my question on MathStackExchange)...
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15 votes
2 answers
2k views

Are there only two smooth manifolds with field structure: real numbers and complex numbers?

Is it true that in the category of connected smooth manifolds equipped with a compatible field structure (all six operations are smooth) there are only two objects (up to isomorphism) - $\mathbb{R}$ ...
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4 votes
0 answers
107 views

Existence of an irreducible polynomial that does not divide $x^n + a$

The question: Can one characterize all fields $K$ over which there exists an irreducible polynomial $f(x)$ that does not divide a polynomial of the form $x^n + a$? Examples: Such a polynomial clearly ...
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41 votes
18 answers
4k views

Results in linear algebra that depend on the choice of field

Linear algebra as we learn it as undergraduates usually holds for any field (even though we usually learn it for the complex, or real, numbers). I am looking for a list of concepts, and results, in ...
1 vote
0 answers
158 views

Why is $\overline{\mathbb{F}_p}((t))$ transcendental over $\mathbb{F}_p((t))$?

Why is $\overline{\mathbb{F}_p}((t))$ transcendental over $\mathbb{F}_p((t))$ ? I guess $\overline{\mathbb{F}_p}((t))$ is not unramified over $\mathbb{F}_p((t))$ because $\overline{\mathbb{F}_p}((t))$ ...
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1 vote
1 answer
232 views

A variation on $k(x^2,x^3)=k(x)$

Let $k$ be a field of characteristic zero, for example $k=\mathbb{R}$ or $k=\mathbb{C}$. Of course, $k(x^2,x^3)=k(x)$, since $x=\frac{x^3}{x^2}$. Let $f_1,\ldots,f_n,g_1,\ldots,g_m \in k[x]$, $n,m \...
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  • 2,573
3 votes
2 answers
229 views

Existence of generic zeros

Let $\Omega$ be an algebraically closed field of characteristic $0$, $k$ a subfield such that $\mathrm{tr.deg}(\Omega/k)=\infty$. Let $u_1,\dots,u_n,u_{n+1}\in \Omega$ be algebraically independent ...
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  • 113
1 vote
0 answers
52 views

Resolvent is minimal polynomial for universal splitting algebra

Given a degree $n$ monic $f\in A[x]$ write $\mathrm{Split}_Af$ for its universal splitting algebra, constructed by taking the quotient of $A[x_1,\dots ,x_n]$ by the Vieta formulas. This is the initial ...
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  • 9,655
3 votes
0 answers
174 views

Simultaneous Galois closure

For a finite separable extension $L/K$ of fields, there exists a Galois closure, which is a finite field extension $\tilde L/L/K$ where $\tilde L/K$ is Galois. (given by the compositum of $\sigma L$, ...
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2 votes
0 answers
138 views

How to decide if an algebraic number is a root of a given polynomial?

Let $p$ be a polynomial with rational coefficients and $\alpha = \sqrt[n]{q}e^{i2k\pi/m}$ for some natural numbers $n,m,k$ and a rational number $q > 0$. Is there an effective algorithm for ...
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3 votes
1 answer
211 views

On tensor product of field extensions

Let $K$ be a field which is a (transcendental) extension of $\mathbb{C}$. Let $L_1, L_2$ and $M_1, M_2$ be two field extensions of $K$ (not necessarily algebraic) such that $$L_1 \otimes_K L_2 \cong ...
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  • 1,533
0 votes
0 answers
106 views

Axiom of Choice and bases of $k$-vector spaces, $k$ fixed

I know that from ZF + the Axiom of Choice (AC) follows that every vector space has a basis. And, conversely, Blass proved that in ZF set theory, the assumption that every vector space has a basis ...
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  • 3,097
1 vote
0 answers
79 views

Separable field extensions and base change

Suppose that there are field extensions \begin{array}{ccc} k & \longrightarrow & K \\ \downarrow & & \downarrow \\ L & \longrightarrow & M \end{array} where $M$ is generated by ...
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  • 435
6 votes
1 answer
282 views

Does solving polynomial equations commute with tropicalization? (particularly for the field of Puiseux series)

The field of Puiseux series over an algebraically closed field of characteristic zero is also an algebraically closed field, and furthermore it has a valuation so that our Puiseux series can be ...
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  • 1,015
1 vote
1 answer
228 views

Product absolute value in rings of integers

Let $F$ be an algebraically closed field of characteristic $p$ equipped with a nonarchimedean dense absolute value $|\cdot|:F \rightarrow \mathbb{R}_{\ge 0}$ with respect to which $F$ is complete. Let ...
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2 votes
0 answers
93 views

Composite of two fields contain a given quadratic extension, but each individual doesn't

In fact, this question could be asked for arbitrary field extension. However, for simplicity I only ask the question for local field of characteristic 0. Let $E/F$ be a quadratic extension of padic ...
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  • 21
6 votes
1 answer
426 views

On the Galois group of the compositions of polynomials

We reprint an old math SE question here (see https://math.stackexchange.com/questions/1241224/composition-of-polynomials-and-galois-theory): " Let $f(x)$ be a polynomial of degree $n$ over $\...
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1 vote
2 answers
267 views

How to show an invariant subfield of rational function field $\mathbb{Q}(x)$ under a certain group action is actually a simple extension? [duplicate]

Let $K=\mathbb{Q}(x)$ be the rational functions in one variable $x$ and let the automorphisms $\phi,\psi$ of $K$ be defined as $\phi(x)=-\frac{1}{x+1}$ and $\psi(x)=\frac{1}{x}$. Let $G$ be the group ...
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5 votes
4 answers
669 views

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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  • 413
2 votes
0 answers
137 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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  • 81
1 vote
2 answers
347 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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7 votes
0 answers
96 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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1 vote
0 answers
50 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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  • 8,348
1 vote
1 answer
422 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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12 votes
0 answers
246 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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2 votes
0 answers
179 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 ...
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  • 53
7 votes
1 answer
557 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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  • 8,348
3 votes
1 answer
179 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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  • 53
1 vote
1 answer
147 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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5 votes
0 answers
163 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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  • 3,097
4 votes
0 answers
191 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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