Maximal subrings up to isomorphism of fields

In this paper, we study maximal subrings up to isomorphism of fields. It is shown that each field with zero characteristic has infinitely many maximal subrings up to isomorphism. If [Formula: see text] is an algebraically closed field, [Formula: see text] is an indeterminate over [Formula: see text] and [Formula: see text] and [Formula: see text] are integrally closed maximal subrings of [Formula: see text] which contain [Formula: see text], then we prove that [Formula: see text]. In particular, if [Formula: see text] is an absolutely algebraic field, then [Formula: see text] has only finitely many integrally closed maximal subrings up to isomorphism if and only if [Formula: see text] is algebraically closed. Also, we show that if [Formula: see text] is an absolutely algebraic field then [Formula: see text] has only finitely many maximal subrings up to isomorphism if and only if [Formula: see text] has only finitely many maximal subrings. We prove that if a commutative ring [Formula: see text] with zero characteristic has only finitely many maximal subrings up to isomorphism, then [Formula: see text] is finite, where [Formula: see text] is the set of natural prime numbers. In particular, if [Formula: see text] is a commutative ring with zero characteristic and [Formula: see text], then [Formula: see text] has infinitely many maximal subrings up to isomorphism. If [Formula: see text] is a field, then we investigate the maximal subrings up to isomorphism of [Formula: see text], [Formula: see text] and [Formula: see text]. If [Formula: see text] is a non-field maximal subring of a field [Formula: see text] and [Formula: see text] is the set of all maximal subrings of [Formula: see text] which are isomorphic to [Formula: see text], then we prove that [Formula: see text], in particular [Formula: see text]. Moreover if, [Formula: see text] is infinite and [Formula: see text], then [Formula: see text] has at least [Formula: see text]-many integrally closed maximal subrings up to isomorphism.