Indépendance algébrique en caractéristique deux

RésuméLet Fq(T)=k, withq=2r, be the rational function field over a finite field of characteristic 2, k∞the algebraic closure of the completion ofkwith respect to the infinite place. The Carlitz exponentiale(z) defined from k∞to itself possesses a kernel generated by (T+Tq)1/(q−1)π, whereπis an analog of the usual number. The reciprocal function ofe(z) denoted Log(z) converges in a neighbourhood of the origin. With this object we prove an analog of the algebraic independence over Q of the two numbers Log(α),αβfor any algebraicαnot equal to zero or one andβquadratic irrational. We also prove, among other things thatπis “hypertranscendental” in the sense thatπandπ′ are algebraically independent. For that purpose, we will construct Drinfeld modules whose derivative of periods are algebraically dependent with the quasi-periods of another Drinfeld module. This property occurs in any characteristicp⩾2