It's actually not that hard. An extension field of field F comes from an irreducible polynomial f(x), irreducible meaning that in any factorization f(x)=g(x)h(x), g or h is a constant. Say f(x)=x^n+cx^{n-1}+...+d. The extension field is then denoted by F[x]/(f(x)), which means that every polynomial with coefficients in F represents an element of the extension field, and you set f(x)=0, so anywhere you see x^n, you can replace it with -(cx^{n-1}+...+d).
For example, the complex numbers are represented in this notation as an extension over the reals R by R[x]/(x^2+1). In this notation, "x" is essentially acting the same as "i" in the conventional complex-number notation, because anywhere you see x^2, you can replace it with -1.
Multiplication works by multiplying polynomials as usual, then reducing the x^n terms to lower-degree polynomials. Inversion of a polynomial g(x) with degree less than n works by solving the equation g(x)h(x)=j(x)f(x)+1. Since g(x) has lower degree than f(x), and f(x) is irreducible, they are coprime, so appropriate h(x) and j(x) can be found using Euclid's algorithm (https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm#S...). OK, maybe that last bit is a little advanced, but Euclid's algorithm is also very simple arithmetic.
Great description. I’d say it’s also important to note that in this context, the division slash means you divide the “numerator” into equivalence classes such that the difference between any two elements in a class is divisible by the “denominator”; i.e., class elements are equivalent mod the denominator. So for instance, x^2 and -1 lie in the same class because x^2-(-1) = x^2 + 1
Yeah, that's important for quotients by more general ideals, but for a principal ideal in a polynomial ring, it's easier to speak of setting the generator to zero.
I banged my head against Chapter 4 for "Algebraic Codes for data Transmission" by Richard E. Blahut. And you need probably Chapter 2 before you can understand Chapter 4. (Chapter 3 is on basic error-correction concepts). The rest of the book is on CRC32, Reed Solomon, and other such error correction concepts. So if you only care about extension fields, its all about Chapter 2 and 4.
I... don't know if I can "recommend" the source, but that's the chapter that finally made me understand extension fields.
Its difficult math. You need to cover groups (number systems that always have "addition"), rings (number systems that always have "addition" and "multiplication"), and fields (number systems that always have "addition", "Multiplication", and "division" ) for... some very precise definition of addition, multiplication, and division.
Once understanding the properties of groups, rings, and fields, the textbook will step you through prime fields. The proof for why prime fields work requires a deep understanding of group and ring properties (so you really can't skip the group / ring discussions).
Then extension fields start to get discussed and the fun really begins. Do you know about polynomials? Such as x^2 - 1 == (x+1) * (x-1) ?? Ever consider that because "x-1" can't be factored, that its kinda-sorta like a prime-polynomial (like prime numbers, a prime polynomial can't be factored).
Ever think about polynomial arithmetic of a polynomial over a modulo of a prime polynomial? Well... there ya go. An extension field. Obviously (/s of course, its not obvious but... that's the gist).
And you know that works because that's pretty similar to arithmetic of a integer over a modulo of a prime integer (aka: the Prime Fields).
Sure, I've got some mathematical training (Fourier transforms, matricies, etc. etc.) but my classes never covered Abstract Algebra. A lot of this stuff is coming out of left-field for me: just never really studied anything in this branch of mathematics.
