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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].

Fields as algebraic objects. For vector and tensor fields, use eg. . For physical fields, use eg. or .

562 questions
22
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4 answers

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 $K_{+} \simeq K_{*} \times \mathbb{Z}/2\mathbb{Z}$ is $K=\mathbb{Z}/2\mathbb{Z}$. Which fields…
Sam Hopkins
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19
votes
1 answer

Is every field the field of fractions of an integral domain?

Is every field the field of fractions of an integral domain which is not itself a field? What about the field of real numbers?
t.k
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16
votes
1 answer

Galois theory: Generalization of Abel’s Theorem? (Better version!)

(Unintentionally I have previously asked a similar and perhaps in itself not uninteresting question Galois theory: Generalization of Abel's Theorem? but this is what I originally had in mind.) Let $L$ stand for the smallest extension of ${\Bbb Q}$…
David Feldman
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13
votes
1 answer

Is every algebraic extension of a field of absolute transcendence degree one a separable extension of a purely inseparable extension?

Any decent course on field theory will state that in characteristic $p$ an extension of fields $k\subset K$ canonically decomposes as the tower $k\subset K_{sep}\subset K$ with $K$ purely inseparable over its subfield $K_{sep}$ of elements…
Georges Elencwajg
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12
votes
2 answers

Why isn't the perfect closure separable?

This question has escalated from math.stackexchange. I'm doing this because it has been a while since the question has been open, receiving no satisfactory answers, even when subject to a bounty. I hope it is adequate for MO, I apologize if it is…
Bruno Stonek
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12
votes
2 answers

Factoring a field extension into one which adds no roots of unity, followed by one which adds only roots of unity

I am asking my question here, since it's been voted up a fair bit on math.SE, but without answers, so it may be harder than I assumed it was. Can we always break an arbitrary field extension $L/K$ into an extension $F/K$ in which the only roots of…
Zev Chonoles
  • 6,722
10
votes
2 answers

uncountable algebraically closed field other than C ?

Is there any "well-known" algebraically closed field that is uncountable other than $\mathbb{C}$ ? The algebraic closure of $\mathbb{C}(X)$ would work, but is it meaningful, is this field used in some topics ? Have you other examples ? Thank you.
Laurent
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10
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3 answers

If K/k is a finite normal extension of fields, is there always an intermediate field F such that F/k is purely inseparable and K/F is separable?

I was feeling a bit rusty on my field theory, and I was reviewing out of McCarthy's excellent book, Algebraic Extensions of Fields. Out of Chapter 1, I was able to work out everything "left to the reader" or omitted except for one corollary, stated…
Zev Chonoles
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7
votes
2 answers

Algebraically closed fields with proper maximal subfields

Is there a classification of the algebraically closed fields that have maximal proper subfields ? And if an algebraically closed field has a maximal proper subfield, is that subfield unique ? Summarizing the answers, an algebraically closed field…
tomasz
  • 557
5
votes
3 answers

Methods of showing an element is / is not in a field

Let $K$ be a field, $\alpha\in\bar{K}$, and $L/K$ a finite extension. How can we determine whether $\alpha\in L$, preferably in as much generality as possible? Of course, there may be special cases where this is easy, e.g. $K\subset\mathbb{R}$ and…
Zev Chonoles
  • 6,722
5
votes
0 answers

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 family of closed subsets, $K$ is a field with an…
Martin Brandenburg
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5
votes
0 answers

Rigid fields containing $\mathbb{C}$

Following the question What is the size of the smallest rigid extension field of the complex numbers?, where it was noted that the least cardinality of a rigid field containing $\mathbb{C}$ is $(2^{\aleph_0})^+$ I have the following question: Is the…
user38200
  • 1,446
3
votes
1 answer

How do you prove that a field is isomorphic to C(x)?

F is a field and F(x) is the field of rational functions in one variable x over F. Are there some clever ways to prove that some field extension K/F is (or is not) isomorphic to F(x)? One way is to try to see if I can embed K into some field which…
iravan
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2
votes
3 answers

Field constructions

If $F$ is a field of characteristic $p$ prime, how can one create a field $K$ such that $K$ is created from $F$ (either by modding out or by taking a product which includes $F$ or by some other method which involves $F$) such that $K$ has a…
user10290
2
votes
2 answers

Making the fixed field algebraically closed

K is a field and f is an automorphism of K. We're also given that the order of f, as an element of the group Aut(K), is not finite. Is it always possible to find a field L which contains K and an automorphism of L, say g, such that g = f over K and…
iravan
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