# Tagged Questions

Use this tag for questions about fields and field theory in abstract algebra. A field is, roughly speaking, an algebraic structure in which addition, subtraction, multiplication, and division of elements are well-defined. Please use (galois-theory) instead for questions specifically about that topic....

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### Homomorphisms between additive and multiplicative groups of fields

Inspired by this question (In a finite field, is there ever a homomorphism from the additive group to the multiplicative group?), I'm wondering for what fields there exists a non-trivial homomorphism ...
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### What branch of math is this?

In this paper: http://arxiv.org/pdf/hep-th/0505016v1.pdf what are the branch(es) of math being used? The unnumbered eq. on the top of page 3 and eq. (7) are good examples. All I've been able to figure ...
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### Learning roadmap in Algebra

I am a senior undergraduate student in mathematics, I have a sound knowledge in the following areas: a) Commutative Algebra b) Field Theory and Galois Theory c) Homological Algebra My question is ...
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### Proving that a certain maximal subfield has countably infinite index

Let $K$ be a maximal subfield of $\mathbb C$ which does not contain $\sqrt{2}$. I've shown that $\mathbb C$ is an algebraic extension of $K$, and that the Galois group of any finite extension of $K$ ...
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### Is there a general algebraic notion of the chain rule?

To motivate this, I should explain that I have been studying differential fields, i.e. fields endowed with a differentiation operator such that $(a+b)'=a'+b'$ and $(ab)'=a'b+ab'$. Using these rules, ...
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### How does the determinant change with respect to a base change?

Problem Suppose $k$ is a (commutative) field, and $A$ is a finite (dimensional) commutative unitary $k$-algebra. $M=A^n$ is a free $A$-module, and therefore can be seen as a finite-dimensional $k$-...
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### A question on Galois theory

In this question, the field of rational numbers is denoted by $\mathbb{Q}$. The Galois group of a polynomial $f$ is denoted by $Gal(f)$. The commutator subgroup of a group $G$ is denoted by $G'$. ...
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### Show that every algebraic field extension has a normal closure

I am trying to show that every algebraic extension of fields has a normal closure. Let $L/K$ be an algebraic extension. First suppose that the extension is finite. So $L=K(\alpha_1,...,\alpha_n)$ ...
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### Question about the index of a subgroup in $\mathrm{Aut}(\mathbb{C} / K )$ with $K$ a number field.

Suppose that $k_0$ is a number field with subfield $K$. Set $[k_0 : K] = d$. If $G = \mathrm{Aut}(\mathbb{C} / K )$ and $H$ is the subgroup of $G$ which fixes $k_0$, is it true that $[G:H] = d$? ...
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### Ordered fields are real?

This question is motivated by another question posted earlier today. Let $F$ be a field endowed with an embedding $F\hookrightarrow\Bbb R$. Then the ordering of $\Bbb R$ induces an ordering on $F$. ...
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### Uses for coefficients other than the trace and norm for an element belonging to a field.

The trace and norm are very useful as maps from a field extension to the base field since they are multiplicative/additive and have a lot of other nice properties. They can be defined as two of the ...
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### Why do fields seem to be a prerequisite for calculus?

I was in my Complex Analysis class, and the professor said that we should look for a field, rather than a group, to do calculus over. Why is this the case? I understand that we gain another operation ...
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### A notion of transcendence degree for fields of formal power series?

There is an intuitive sense in which one would like to say that the field of formal Laurent series $k((z))$ over a field $k$ has "transcendence degree $1$". Of course, it doesn't have transcendence ...
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### The existence of Pisot numbers in any real number field

Wikipedia claims that, given a real algebraic number field $K$ of degree $n$, there is an algebraic integer $r \in K$ of degree $n$ such that $r>1$, but every conjugate of $r$ has modulus $<1$ (...