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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!

edited Nov 12 2009 at 16:44

Scott Morrison♦
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asked Oct 23 2009 at 23:10

Andrew Critch
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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. – Regenbogen Feb 26 2010 at 0:15

Sonia Balagopalan · Accepted Answer · 2009-11-09 15:16:41Z UTC

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Finite Fields by R. Lidl and H Niederreiter (CUP). Probably as comprehensive as it gets.
The ams review calls it the ``the Bible of finite fields''. You can find it (the review)here.

edited Nov 9 2009 at 15:16

answered Oct 23 2009 at 23:44

Sonia Balagopalan
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Your link appears to be broken. – Alison Miller Oct 24 2009 at 0:37

Fixed it now. Thanks! – Sonia Balagopalan Oct 24 2009 at 0:58

It's great, thank-you! – Andrew Critch Oct 24 2009 at 5:49

Ilya Nikokoshev · Answer 2 · 2009-10-23 23:14:24Z UTC

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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.

answered Oct 23 2009 at 23:14

Ilya Nikokoshev
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What I'm looking for is a single comprehensive source (just edited the question to reflect this). – Andrew Critch Oct 23 2009 at 23:25

A comprehensive overview of finite fields

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