The purpose of the paper is to give the state of the art on Michael's problem, the long-standing open question odf continuity of characters on commutative Fréchet algebras. We first quote two well-known consequences of the "abstract Mittag-Leffler theorem", the theorem of Arens, which shows that characters on finitely rationally generated Fréchet algebras are continuous, and the fact that the existence of a nonincreasing sequence (Ωn) n≥1 of Fatou-Bieberbach domains in C p such that ∩ n≥1 Ωn = ∅ would imply that all characters on commutative Fréchet algebras are continuous. In the opposite direction the existence of a discontinuous character on some commutative unital Fréchet algebra is equivalent to the existence of a character on a quotient algebra of the form U/I where U is a 'test algebra' for Michael's problem and where I is a dense ideal of U which is a Picard-Borel ideal, which means that every family of pairwise linearly independent invertible elements of U/I is linearly independent. It was recently shown that all Picard-Borel ideals in commutative unital Fréchet algebras are prime, and Picard-Borel ideals of H(C) can be easily described. We raise a question concerning Picard-Borel ideals of H(C p), p ≥ 2 which could lead to important general information about the quotient of commutative unital Fréchet algebras by Picard-Borel ideals. The fact that entire functions of several variables oerate on quotients of Fréchet algebras by ideals which are not necessarily closed plays an essential role in the paper.< Leer menos