GAP Manual: 13.10. Operations for Cyclotomics

The operators +, -, *, / are used for addition, subtraction, multiplication and division of two cyclotomics; note that division by 0 causes an error.

+ and - can also be used as unary operators;

^ is used for exponentiation of a cyclotomic with an integer; this is in general not equal to Galois conjugation.

    gap> E(5) + E(3); (E(5) + E(5)^4) ^ 2; E(5) / E(3); E(5) * E(3);
    -E(15)^2-2*E(15)^8-E(15)^11-E(15)^13-E(15)^14
    -2*E(5)-E(5)^2-E(5)^3-2*E(5)^4
    E(15)^13
    E(15)^8