Some Maximal Function Fields and Additive Polynomials

Abstract

We derive explicit equations for the maximal function fields F over 𝔽 _{q ²ⁿ} given by F = 𝔽 _{q ²ⁿ} (X, Y) with the relation A(Y) = f(X), where A(Y) and f(X) are polynomials with coefficients in the finite field 𝔽 _{q ²ⁿ} , and where A(Y) is q-additive and deg(f) = q ⁿ  + 1. We prove in particular that such maximal function fields F are Galois subfields of the Hermitian function field H over 𝔽 _{q ²ⁿ} (i.e., the extension H/F is Galois).

Key Words:

• Additive polynomial
• Finite field
• Maximal curve

2000 Mathematics Subject Classification:

• Primary 11G20, 14G05
• Secondary 14G50

ACKNOWLEDGMENTS

The first author acknowledges partial support from CNPq-Brazil (grant 304628/ 2003-4) and from PRONEX (CNPq-FAPERJ).

The second author acknowledges partial support from the Turkish Academy of Sciences in the framework of the Young Scientists Award Programme (F.Ö./TÜBA-GEBIP/2003-13).

Notes

Communicated by S. Kleiman.