We study totally real number fields that admit a universal quadratic form whose coefficients are rational integers. show Q(5) is the only such field, and among of degrees 3, 4, 5, 7 which have principal codifferent ideal, one Q(ζ7+ζ7−1), over x2+y2+z2+w2+xy+xz+xw universal. Moreover, we prove an upper bound for Pythagoras numbers orders in depends on degree field.

Let B=A?X|dX=t? be an extended DG algebra by the adjunction of a variable positive even degree n, and let N semi-free B-module that is assumed to bounded below as graded module. We prove in this paper liftable A if ExtBn+1(N,N)=0. Furthermore such lifting unique up isomorphisms ExtBn(N,N)=0.

(1b) Question. For which pairs (C0,H) does a lift exist? Note that the lifting problem for C0 is formally smooth. However we will see that in general the lifting problem for (C0,H) can be obstructed; in some cases a lifting does not exist, in several cases ramification in R is needed to make a lifting possible. In order to have a positive answer to this question it suffices to consider the case...