  We classify all totally real number fields of degree at most 5 that admit a
universal quadratic form with rational integer coefficients; in fact, there are
none over the previously unsolved cases of quartic and quintic fields. This
fully settles the lifting problem for universal forms in degrees at most 5. The
main tool behind the proof is a computationally intense classification of
fields in which every multiple of 2 is the sum of squares. We further extend
these results to some real cyclotomic fields of large degrees and prove
Kitaoka's conjecture for them.
