(Finite) Fields — A Primer

Math ∩ Programming

So far on this blog we’ve given some introductory notes on a few kinds of algebraic structures in mathematics (most notably groups and rings, but also monoids). Fields are the next natural step in the progression.

If the reader is comfortable with rings, then a field is extremely simple to describe: they’re just commutative rings with 0 and 1, where every nonzero element has a multiplicative inverse. We’ll give a list of all of the properties that go into this “simple” definition in a moment, but an even more simple way to describe a field is as a place where “arithmetic makes sense.” That is, you get operations for $latex +,-, cdot , /$ which satisfy the expected properties of addition, subtraction, multiplication, and division. So whatever the objects in your field are (and sometimes they are quite weird objects), they behave like usual numbers in a very…

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