$\mathbf{E}$-compact extensions in the absence of the Axiom of Choice

The main aim of this work is to show, in the absence of the Axiom of Choice, fundamental results on $\mathbf{E}$-compact extensions of $\mathbf{E}$-completely regular spaces, in particular, on Hewitt realcompactifications and Banaschewski compactifications. Some original results concern a special subring of the ring of all continuous real functions on a given zero-dimensional $T_1$-space. New facts about $P$-spaces, Baire topologies and $G_{\delta}$-topologies are also shown. Not all statements investigated here have proofs in $\mathbf{ZF}$. Some statements are shown equivalent to the Boolean Prime ideal Theorem, some are consequences of the Axiom of Countable Multiple Choices.