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An elementary proof using matrix theory is given for the following criterion: if F/K and L/K are field extensions, with F and L both contained in a common extension field, then F and L are linearly disjoint over K if (and only if) some K-vector space basis of F is linearly independent over L. The material in this note could serve as enrichment material for the unit on fields in a first course on abstract algebra. [ABSTRACT FROM AUTHOR]

Let F be a global field with ring of integers A, and let K ⊆ L be a finite-dimensional Galois extension of fields algebraic over F, with B (resp., C) the integral closure of A in K (resp., L). If a nonzero prime ideal P of B has only one prime ideal 𝔓 of C lying over it and char(B/P) does not divide [L: K], then Gal(L/K) is a metacyclic group. Thus, if 0 < char(F) does not divide [L: K] and Gal(L/K) has a minimal generating set of cardinality at least 3, then each nonzero prime ideal P of B has at least two distinct prime ideals of C that lie over it. For any odd prime number p, an example is given of the latter situation where char(F) = p. [ABSTRACT FROM AUTHOR]