> I usually explain extension fields as similar to complex numbers with regards to reals.
Its a good analogy and the math is indeed very close to complex vs reals.
If you consider "i" to be the "polynomial variable", then a prime-integer field vs a (prime-integer)^2 extension field is pretty much identical to real vs complex.
Of course, there's a prime^3 extension field, or a prime^4 extension field, and then the analogy kind of stops working.
EDIT: Now that I think of it... I can't really decide if complex-numbers are like a prime^4 field or like a prime^2 field. i^4 == 1 after all.
In a prime^2 field, the variable x^2 == 1. In a prime^200 field, x^200 == 1. Etc. etc.
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Really, the glorious thing about extension fields is that a well selected extension field follows the fundamental theorem of algebra (which is... in simple terms... "All Polynomial equations have an answer that can be represented by your number system". Ex: (x^2 + 1 == 0) not only can be solved by a complex number x = i, but all such possible equations are proven to have an answer).
So you get all the benefits of integers (such as perfectly mapping to 2^8 == 256), with none of the downsides of real (aka: uncountably infinite, rounding errors, etc. etc.). You get precision while still keeping your property of "having answers" to a huge class of important algebraic problems.