When working with finite fields, if the number of elements is a prime power with m > 1, you can represent the elements as polynomials with degree m-1 and do the field addition and multiplication modulo a irreducible polynomial with degree m.
The field GF(5) is composed by the numbers 0 to 4. We don’t need to represent its elements as polynomials since m=1. Addition is done modulo 5 and multiplication also modulo 5. So 2 + 3 = 0; 4 * 2 = 3; and so on. This is the addition table for GF(5):
The rows, top down, represent 0 to 4. The columns, right to left, represent 0 to 4. Each square is the result of the addition of the respective numbers in the row / column it belongs to. Black is 0, purple is 1, red is 2, orange is 3, yellow is 4.