# 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 ...

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### $\mathbb{C^2}$ treated as a real vector space

As a real vector space the dimension is 4. What is an orthonormal basis for it with respect to either the standard real inner product?. I've tried gram schmidt with the obvious choice of 4 vectors( ...
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### Uniqueness of the real line

A few days ago, I came across this question in a review queue. I tried my luck at it. Here is what I did: If I want a homomorphism (isomorphism, but even just homomorphism) $f:\mathbb{R}\to F$, then ...
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### Constructing a field so that a polynomial has a root.

This was on my study guide given out by my professor. I could not find this anywhere in my book though. Any advice on how to do this. I understand the concept that a polynomial may not have a root ...
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### Field extension is étale implies polynomial is separable

Following Johnstone (Exercise 0.11), a ring homomorphism $f: A\rightarrow B$ is étale if for every nilpotent ideal $N\subseteq R$ of a ring $R$ and every diagram of ring homomorphisms there is a ...
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### Product of elements in finite field

Let $q$ be a prime or primer power such that $\mathbb{F}_q$ is a finite field. Now consider the extension $\mathbb{F}_{q^m}$, which may be regarded as a vector space of dimension $m$ over ...
I'm working on a question saying $A \subseteq B \subseteq C$ fields, and $a \in C$ algebraic over $A$, either prove $[A(a):A] ≥ [B(a):B]$ or give a counter example. I think it's true and used the ...
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, ...