Reply To: !
@upsidedown Yes, well done. As Harry said, finite fields are very useful and a pretty cool things to study. An explicit construction involves writing it as a quotient ring. A quotient you can think of as a generalisation of modular arithmetic. To get the integers modulo 5 (Z/5Z) we start with the integers and enforce the condition that things which differ by a multiple of 5 (e.g. 11 and 1) are the same. Similarly we can form a quotient of polynomial rings: for instance we can take the set of all real polynomials and make the identification that things which differ by a multiple of x^2 + 1 are the same (e.g. x^2 + 2 and 1 or x^3+2x+1 and x+1). By the division algorithm, everything in this quotient can be written as a + bx for some a, b in R. One can see that this quotient is equivalent to the complex numbers (you can think of this as extending the reals with a root of the polynomial x^2 + 1). To construct finite fields one can do a similar thing where we quotient GF(p)[x] (polynomials in the integers mod p) by an irreducible polynomial. This is a very brief description and skips over most of the technical details, so it is okay if this doesn’t make any sense.
@Harry, I would be interested to hear your recommendations for textbooks on more modern cryptography. I have read Rational Points on Elliptic Curves up to the section on Mordell’s theorem, but this is at a rather basic level. How much modern Algebraic geometry is needed for this stuff, because frankly a lot of that scares me; it seems everyone I run into who works on it is a total genius whose bedtime reading as a child was Bourbaki’s commutative algebra.