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8
votes
1answer
578 views

Existence of maximal totally ramified extensions of an arbitrary CDVF

Let $K$ be a complete, discretely valued field with (let's say) perfect residue field $k$. We have a unique maximal unramified extension $K^{unr}$ of $K$ and a unique maximal tamely ramified ...
117
votes
0answers
10k views

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 ...
20
votes
0answers
948 views

Subfields of $\mathbb{C}$ isomorphic to $\mathbb{R}$ that have Baire property

While sitting through my complex analysis class, beginning with a very low level introduction, the teacher mentioned the obvious subfield of $\mathbb{C}$ isomorphic to $\mathbb{R}$, and I then ...
16
votes
0answers
382 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 ...
15
votes
0answers
307 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 ...
14
votes
0answers
685 views

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. ...
7
votes
0answers
216 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? ...
5
votes
0answers
249 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} ...
5
votes
0answers
322 views

Point of confusion in “Topological Representations of Algebras”

Background I'm reading the article "Topological Representations of Algebras" by Arens, Kaplansky. In the proof of Theorem 6.1 we have the following situation: $X$ is a Stone space, $X_\alpha$ is a ...
4
votes
0answers
350 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
0answers
131 views

Analogue of a ring extension splitting in the Kummer case

Background (the Kummer extension case) Let $R$ be a complete regular local ring (it follws that it's a UFD) with a prime integer $p$ contained in the maximal ideal of $R$ (I'm mostly interested ...
2
votes
0answers
59 views

Extension of the product formula for valuations to a simultaneous completion

It is well known that $\mathbb{C}$ and $\mathbb{C}_p$ are "algebraically" isomorphic (that is, ignoring the topology), but an isomorphism depends on the axiom of choice and there is no canonical way ...
2
votes
0answers
71 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
0answers
139 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 $k$?
2
votes
0answers
107 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 ...
2
votes
0answers
193 views

is exponential diophantine over Qp

Thanks to Matiyasevic, we all know that exponential is diophantine over the integers. Also, thanks to transcendental number theory, we know that exponential is not diophantine over the rationals. So ...
2
votes
0answers
353 views

What do you call an algebraic element with the property that the generated field extension is normal?

Let $L/K$ be a field extension. Let $\alpha \in L$ be algebraic over $K$. Is there an established terminology for the property of $\alpha$ that $K(\alpha)/K$ is a normal field extension? Would you ...
1
vote
0answers
107 views

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 ...
1
vote
0answers
110 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 + ...
1
vote
0answers
33 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, ...
1
vote
0answers
119 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 ...
1
vote
0answers
64 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 ...
1
vote
0answers
96 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 ...
1
vote
0answers
88 views

when is the property “being algebraically maximal” a first order property ?

A valued field is said to be algebraic maximal if all its algebraic extension have either a bigger value group or a bigger residue field. Do you know for which field this is a first order property ? ...
1
vote
0answers
150 views

Extending commuting endomorphims of a complete discrete valued field to the algebraic closure?

Is it true that any two commuting endomorphisms of a complete discrete valued field extend to commuting automorphisms of the algebraic closure?
1
vote
0answers
208 views

Automorphism groups of fields

Hi there, is there a classification/characterization of fields K for which the automorphism group Aut(K) has the property that |Aut(K)| < |K| (e.g. finite fields, the rationals and reals) ? What ...
1
vote
0answers
79 views

Explicit expression for multivariable meromorphic series

Let $K$ be a field. For $m>1$, elements in the ring of power series in $m$ variables over $K$, i.e., $K[[X_1, \cdots, X_m]]$ look like - $$ \displaystyle\sum_{(i_1, \cdots, i_m)\in (\mathbb ...
1
vote
0answers
343 views

Quadratic Solutions

There are quadratic solutions to $x^4+y^4 = z^4$ in $\mathbb{Q} (\sqrt{-7})$. But for equations such as $x^4+y^4 = nz^4$ where $n \in \mathbb{N}, \ n \neq 1$ do there still exist extension fields of ...
0
votes
0answers
69 views

Algorithm for finding irreducible polynomials in finite field extensions

Let $K(\alpha_1,\ldots,\alpha_n)/K=\tilde{K}/K$ be a finite field extension and suppose we know $\text{irr}(\alpha_1,K)(x),\ldots,\text{irr}(\alpha_n,K)(x)\in K[x]$. Suppose also that we have a basis ...
0
votes
0answers
144 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 ...
0
votes
0answers
157 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 + ...
0
votes
0answers
119 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 $\mathbb{Q}_{p}$?
0
votes
0answers
423 views

Why do we need to show that two irreducible polynomials divide each other?

Suppose we have the conjugation isomorphism $\psi_{\alpha, \beta}: F(\alpha) \mapsto F(\beta)$ defined by $$\psi_{\alpha, \beta}(a_0+a_{1} \alpha + \cdots + a_{n-1} \alpha^{n-1}) = a_0+a_{1} \beta + ...