Fp×

Fp× denotes the multiplicative group of nonzero elements of the finite field Fp, where p is a prime. The field Fp consists of the p integers {0, 1, ..., p−1} with addition and multiplication performed modulo p. The set Fp× = Fp \ {0}, equipped with multiplication modulo p, forms a finite abelian group of order p−1. It is a cyclic group, meaning there exists an element g in Fp× (a primitive root) such that every nonzero element of Fp can be written as g^k for some k with 0 ≤ k ≤ p−2. Consequently, every element has a multiplicative inverse, and the group is isomorphic to the cyclic group of order p−1.

Notation: Fp× is sometimes written Fp* or (Fp)×.

Properties and consequences: For any a in Fp×, the order of a divides p−1 (Lagrange’s theorem). In particular, a^(p−1) ≡ 1 mod p (Fermat’s little theorem). The case p = 2 yields F2× = {1}, a trivial group. Primitive roots exist for every prime p, establishing the cyclic structure of Fp×.

Examples: For p = 7, F7× = {1, 2, 3, 4, 5, 6}, and 3 is a primitive root since its powers mod 7 cycle through all six nonzero residues. For p = 5, F5× = {1, 2, 3, 4} with 2 as a primitive root.

Extensions: The concept generalizes to finite fields F_{p^n} where F_{p^n}× is cyclic of order p^n−1. Fp× is central in modular arithmetic and number theory, including discrete logarithms and primitive-root computations.