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A quadratic form over a non-archimedian local field of characteristic zero$F$ is called universal if it is integral and it represents all non-zerointegers of $F$. Xu Fei and Zhang Yang determined all universal quadratic formsin the case when $F$ is non-dyadic. In the more complicated dyadic case, when$F$ is a finite extension of $\mathbb Q_2$, they solved the same problem onlyin the ternary case. In our paper we solve this problem in the general case.Our result is given in terms of BONGs (bases of norm generators) but in section3 of the paper we translate (without a proof) our result in terms of the moretraditional Jordan splittings.  In the last section we give some results on $n$-universality. We show that itcan be reduced to the cases $n\leq 4$ and we give explicit necessary conditionsfor $n$-universality in the case when $n\geq 3$, $n$ odd. (A quadratic form iscalled $n$-maximal if it is integral and it represents all non-degenerateintegral quadratic forms of rank $n$.)

Universal integral quadratic forms over dyadic local fields