Some remarks on subfields of algebraic number fields (Q1925166)

In this note it is proved that if \(k\) is an algebraic number field, for every positive integer \(n\), there exist infinitely many field extensions of \(k\) of degree \(n\) having no proper subfields over \(k\). This result is deduced from the existence of infinitely many polynomials of degree \(n\) over \(k\) with Galois group \(S_n\) and of no proper subgroups between \(S_n\) and \(S_{n-1}\).