GFq2

GFq2 is a finite field, specifically a Galois field, with 2^2 elements. It is a fundamental structure in abstract algebra and has applications in various fields such as coding theory, cryptography, and computer science. The field GFq2 is constructed using polynomials over the binary field GF2, which has only two elements: 0 and 1. The elements of GFq2 can be represented as binary polynomials of degree less than 2, namely 0, 1, x, and x+1, where x is a root of the irreducible polynomial x^2 + x + 1.

The arithmetic operations in GFq2 are performed modulo the irreducible polynomial x^2 + x + 1. Addition is simply bitwise XOR, while multiplication is polynomial multiplication followed by reduction modulo the irreducible polynomial. The field GFq2 is a cyclic group under multiplication, and its multiplicative group has order 3, generated by the element x.

GFq2 is a small finite field, but it serves as a building block for larger finite fields, such as GFq4, GFq8, and so on. These larger fields are constructed using extensions of GFq2 and share similar properties. The study of finite fields, including GFq2, is a rich area of research with numerous theoretical and practical implications.