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I've read numerous introductions to finite fields, but I feel like my intuition about them is fairly lacking. Considering that finite fields are the the most "inert" objects in algebraic geometry, I think I could use a serious surge of perspective.

What I would like to read now is a comprehensive overview that tells me "everything I need to know" about how finite fields and their algebraic closures work, algebraically. I don't mind working out the proofs on my own if they are terse or absent; I'm just looking for quality and quantity of results. Hopefully some intense reading will help steep out some of my insecurities about characteristic p.

Can anyone recommend a single source for such an overview?

Thanks!

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  • $\begingroup$ If you are a geometer over finite fields, then the Frobenius will make sure that you are almost as good as in the algebraically closed case. Befriend the Frobenius, and then you are in a safe position. $\endgroup$
    – Regenbogen
    Commented Feb 26, 2010 at 0:15

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Finite Fields by R. Lidl and H Niederreiter (CUP). Probably as comprehensive as it gets (see also Google Books). Gary Mullen's Bull. AMS review calls it the "the Bible of finite fields".

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The really important things in algebraic number theory start from group cohomology and theorems like Hilbert 90, but you'll be better searching/asking for different keywords than finite fields then.

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  • $\begingroup$ What I'm looking for is a single comprehensive source (just edited the question to reflect this). $\endgroup$
    – Andrew Critch
    Commented Oct 23, 2009 at 23:25
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The "Handbook of Finite Fields" is probably the most comprehensive book, treating all aspects of finite fields and their applications.

Mullen, G.L., & Panario, D. (2013). Handbook of Finite Fields (1st ed.). Chapman and Hall/CRC. https://doi.org/10.1201/b15006

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