Galois (finite) fields are very much relevant in error correction codes and cryptography. Any time you want to manipulate bytes (or words) as numbers which you can add/subtract and multiply/divide you end up learning about finite fields. In fact, doing CRC or even XOR sums you are already making the first step towards the topic.

A recent problem I needed a solution for: given n blocks of data of equal length generate all possible XOR sums of the blocks in-place (no additional memory). The answer is an irreducible polynomial of degree n with a primitive root over GF(2).