symmetric monoidal (∞,1)-category of spectra
A number field is a finite field extension of the field of rational numbers, $\mathbb{Q}$, In other words, a field $k$ of characteristic zero such that under the field homomorphism $i: \mathbb{Q} \hookrightarrow k$, the field $k$ is a finite-dimensional vector space over $\mathbb{Q}$ with respect to the scalar multiplication action of $\mathbb{Q}$
on the underlying additive group of $k$.
the rational numbers $\mathbb{Q}$;
the Gaussian numbers $\mathbb{Q}(i)$;
for certain $d$ the quadratic field? $\mathbb{Q}(\sqrt{d})$;
the cyclotomic fields $\mathbb{Q}(\zeta_n)$
Counterexamples:
Number fields are the basic objects of study in algebraic number theory. For example, one is typically interested in the arithmetic structure of $k$, including for example the structure of the ring of algebraic integers $\mathcal{O}_k$ in $k$, the decomposition of primes in $\mathbb{Z}$ in terms of prime ideals in $\mathcal{O}_k$, the structure of the unit group of $\mathcal{O}_k$, the structure of the ideal class group?, the detailed study of the zeta function of $k$, and much more.
Number fields $k$ are examples of global field?s, in fact they are the global fields of characteristic zero. They are often studied in terms of how they embed in their rings of adeles $\mathbb{A}_k$, which are built from the local completions of $k$.