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My goal is to fully understand this answer on crypto.stackexchange by self-teaching myself all the basics.

The term I'm working on now is a "Galois field", and starting on this wiki page. The header of a page has a header like this:

Commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ finite fields.

Which makes me believe that either Commutative rings is a subset of finite fields. (or the other way around, I don't know yet.)

What is the best learning path I should start on to understand finite fields?

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makerofthings7
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    Well, "commutative rings" is a huge topic, you don't so much "learn" them as study them. But it is helpful to know a little about commutative rings (and about abelian groups) before learning finite fields. – Thomas Andrews Jul 19 '13 at 17:43
  • And it is the other way around. Every finite field is a commutative ring. – Thomas Andrews Jul 19 '13 at 17:46
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    There's not much depth in that crypto.SE answer - it just says that modular arithmetic is used. The fact that integers modulo a prime form a finite field is a very small part of the topic of finite fields, and should be able to be found in most elementary/introductory number theory books. There are other uses for finite fields in cryptography, if I recall correctly, but that answer doesn't mention them. In particular, you don't need to know what a Galois field is, really, to understand that question - it's just a fancy term for generalizations of the integers modulo $p$. – Thomas Andrews Jul 19 '13 at 18:11
  • Most textbooks use commutative rings (well, the special case of a polynomial ring in one variable) to explicitly construct finite fields. I'd learn some basics of polynomial rings first if you want to understand that. – Matt Jul 19 '13 at 19:10
  • Yeah. Something titled approximately "A first course in abstract algebra" will get you started (do check the contents it will go up to polynomial rings and ideals). Such a tome may or may not get to finite fields, but reading it will make it easier to grasp finite fields. There are three basic theorems about finite fields: their existence, uniqueness up to isomorphism and cyclicity of the multiplicative group. Depending on what you want to do with them, you may or may not need to understand the proofs of the listed results. – Jyrki Lahtonen Jul 19 '13 at 20:06
  • cont'd ... But if you are serious about this, then you need to learn basics of field theory. Otherwise trying to get a handle on stuff like reducing operations of $GF(256)$ (i.e. AES) to operations of $GF(16)$ for more efficient hardware implementations will all be Greek to you. The papers linked to in this question being a case in point. Of course, very large finite fields are used in e.g. elliptic curve crypto. There the underlying algebra runs a lot deeper. – Jyrki Lahtonen Jul 19 '13 at 20:13

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