  We show that if $K$ is a monogenic, primitive, totally real number field,
that contains units of every signature, then there exists a lower bound for the
rank of integer universal quadratic forms defined over $K$. In particular, we
extend the work of Blomer and Kala, to show that there exist infinitely many
totally real cubic number fields that do not have a universal quadratic form of
a given rank defined over them. For the real quadratic number fields with a
unit of negative norm, we show that the minimal rank of a universal quadratic
form goes to infinity as the discriminant of the number field grows. These
results follow from the study of interlacing polynomials. Specifically, we show
that there are only finitely many irreducible monic polynomials related to
primitive number fields of a given degree, that have a bounded number of
interlacing polynomials.
