Resources for concrete understanding of fields

Question

I'm looking for a textbook that provides exercises for developing intuition about fields themselves, rather than extentions. For example I can't really picture in my head what $\bar{\mathbb{Q}}$ or $\bar{\mathbb{Z}_p}$ 'look like'.

Answer 1

Extensions are definitely one of the most important tools for understanding fields - they describe how one field is contained in another. Avoiding them would be like saying that you want to get an intuition for each individual part of a car, but don't want to think about how they fit together. For a famous exammple, when we talk about Galois groups, we talk about the Galois group of an extension, $\mathrm{Gal}(L/K)$. And the algebraic closure $\overline{\mathbb{F}_p}$ of the finite field with $p$ elements $\mathbb{F}_p$ is just the union of all the finite extensions $\mathbb{F}_{p^n}$ of $\mathbb{F}_p$, so understanding them and how they fit together is very important - in particular, that $\mathbb{F}_{p^n}$ is a subfield of $\mathbb{F}_{p^m}$ if and only if $n\mid m$.

Moreover, since any ring homomorphism from one field to another is necessarily injective, avoiding the study of extensions is practically the same as avoiding homomorphisms! Surely you can agree that they are important.

In terms of books that cover the general structure theory of fields, and have exercises, I would suggest:

• Field Arithmetic by Michael D. Fried, Moshe Jarden

• Algebra: Volume 1 and Algebra: Volume II by Falko Lorenz

• Algebraic Extensions of Fields by Paul J. McCarthy

Another book without exercises is

• Field Theory by Gregory Karpilovsky