On divisibility of the class number $h^{+}$ of the real cyclotomic fields of prime degree $l$

In this paper, criteria of divisibility of the class number $h^+$ of the real cyclotomic field $\mathbf{Q}(\zeta _p+\zeta _p^{-1})$ of a prime conductor $p$ and of a prime degree $l$ by primes $q$ the order modulo $l$ of which is $\frac{l-1}{2}$, are given. A corollary of these criteria is the possibility to make a computational proof that a given $q$ does not divide $h^+$ for any $p$ (conductor) such that both $\frac{p-1}{2},\frac{p-3}{4}$ are primes. Note that on the basis of Schinzel's hypothesis there are infinitely many such primes $p$.