Fp2

Fp2, also written GF(p^2), denotes the finite field with p^2 elements, where p is a prime. It is the quadratic extension of Fp and is unique up to isomorphism. A standard construction is Fp[x]/(x^2 − u), where u ∈ Fp is a non-square. Let α be the class of x; then α^2 = u and every element can be written as a + bα with a,b ∈ Fp. Addition is coordinate-wise, and multiplication uses α^2 = u: (a + bα)(c + dα) = (ac + bdu) + (ad + bc)α.

The field has characteristic p and contains Fp as a subfield. Its multiplicative group Fp^2× has order p^2 − 1 and is cyclic. The Frobenius automorphism φ: x ↦ x^p generates the Galois group of the extension and satisfies α^p = −α. The trace and norm from Fp^2 to Fp are Tr(a + bα) = 2a and N(a + bα) = a^2 − b^2u, respectively.

For p = 2 a common choice is the irreducible polynomial x^2 + x + 1 over F2, yielding GF(4). In general, any irreducible quadratic over Fp yields a model of Fp^2.

Fp2 is widely used in number theory and cryptography as an extension field over which arithmetic is performed, including in elliptic curves, pairing-based protocols, and various algorithms requiring a small, well-structured finite field. It is the standard finite field of order p^2 and is unique up to isomorphism.