Non-universality of ternary quadratic forms over fields containing $\sqrt{2}$

We prove Kitaoka’s conjecture for all totally real number fields of degree 4 – namely, there is no positive definite classical quadratic form in three variables which is universal. To achieve this, we study the fields (often without restricting the degree) where 2 is a square, because in this arguably most difficult case, the recent results connecting Kitaoka’s conjecture to sums of integral squares do not apply. We also prove some other properties of ternary quadratic forms over fields containing $\sqrt{2}$, for example in relation to the lifting problems for universal quadratic forms and for criterion sets.