Short answer: This is open in general. There is no known theorem that every number field K embeds in a number field L whose ring of integers OL is generated by one element over Z; nor is there a known counterexample.

Remarks and partial results:

• Positive cases:

  • If K is abelian over Q, then by Kronecker–Weber there exists m with K ⊂ Q(ζm), and cyclotomic fields are monogenic: OQ(ζm) = Z[ζm]. Thus the answer is yes for abelian K (take L = Q(ζm)).
  • Quadratic fields are themselves monogenic, so the question is trivial there.

• Why the general case is hard:

  • Monogenicity is quite restrictive and is not preserved under common field operations (e.g., composita of monogenic fields need not be monogenic).
  • Relative monogenicity (OL = OK[α]) does not imply absolute monogenicity (OL = Z[β]).
  • Non-monogenicity can often be detected by local obstructions (via index considerations at primes), but it is not known how to rule out monogenicity for all extensions L/K of a given K.

• References:

  • The existence and finiteness of power integral bases are surveyed in K. Győry’s work and in the monograph by J.-H. Evertse and K. Győry, Unit Equations in Diophantine Number Theory (Cambridge, 2015). They discuss the state of the art and list as an open problem whether every number field embeds in a monogenic number field.
  • See also A. Gaál, Diophantine Equations and Power Integral Bases (Birkhäuser, 2002), for background and further references.

So, while many concrete families of L are known to be monogenic (e.g., cyclotomic fields and certain “pure” fields), and while one can often produce monogenic extensions in special situations, it is currently unknown whether every number field sits inside some finite extension with a power basis for its ring of integers.