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A term is in beta-normal form when it contains no beta-redex, meaning no subterm of the formEquivalently, no beta reduction can be performed anywhere in the term.

The Weak normalization theorem for simply typed lambda calculus states that every well-typed term of the simply typed lambda calculus admits at least one finite sequence of beta reductions ending in a beta-normal form.

No. The Omega combinator isIts only beta-redex contracts back to $\Omega$ itself. Every reduction sequence therefore repeats the same term, which is not in beta-normal form. Thus $\Omega$ has no beta-normal form.

Suppose a fixed-point combinator $F$ were typable in the simply typed lambda calculus. By the Weak normalization theorem for simply typed lambda calculus, it would have a beta-normal form. For a fresh variable $f$, the term $Ff$ would then also possess a beta-normal form, say $N$.

The fixed-point property givesReducing the occurrence of $Ff$ on the right to $N$ gives the normal form $fN$. The Church-Rosser theorem says that these beta-equivalent terms must have alpha-equivalent normal forms. This is impossible because $fN$ contains more symbols than $N$. Hence no typing context and simple type can type $F$.