Suppose $k$ is a number field, i.e. an extension of $\mathbb{Q}$ of finite degree, so we have a natural inclusion $\mathbb{Q} \rightarrow k$, which induces a morphism,
\begin{equation}
\text{Spec}\,k \rightarrow \text{Spec}\,\mathbb{Q}
\end{equation}


I have a naive (probably wrong) question to bother the mathoverflow community: is there a (mixed)-motive (over $\mathbb{Q}$ with coefficient ring $\mathbb{Q}$ ) associated to the $\mathbb{Q}$-scheme $\text{Spec}\,k$? If so, what is its Betti ($\ell$-adic) realisation? Please forgive my naiveness.



Motives associated to a Number Field
Write $X=\mathrm{Spec}\, k$, which is a $0$-dimensional variety. Motives of $0$-dimensional varieties are called Artin motives, and they are pure. The Betti realization is the Betti cohomology of $X$, which is
$$
H^0(X(\mathbb{C}),\mathbb{Q})=\mathbb{Q}^{X(\mathbb{C})}=\mathbb{Q}^{\mathrm{Hom}(k,\mathbb{C})}.
$$
There is an involution induced by complex conjugation on $\mathbb{C}$.


To define the $\ell$-adic realization, we need to choose an algebraic closure $\overline{\mathbb{Q}}$ of $\mathbb{Q}$ (of course the one inside $\mathbb{C}$ is a common choice). The basechange $X\_{\overline{\mathbb{Q}}}:=X\times\_{\mathrm{Spec}\,\mathbb{Q}}\mathrm{Spec}\,\overline{\mathbb{Q}}$ can be identified with the set $\mathrm{Hom}(k,\overline{\mathbb{Q}})$ (viewed as a constant scheme over $\overline{\mathbb{Q}}$), and the $\ell$-adic realization is then
$$
H^0\_{et}(X\_{\overline{\mathbb{Q}}},\mathbb{Q}\_\ell)=\mathbb{Q}\_{\ell}^{\mathrm{Hom}(k,\overline{\mathbb{Q}})}.
$$
This space comes with an action of $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$.


Artin motives are discussed in Section 9.4 of [1] and Section 6 of [2].


[1] *Huber, Annette; Müller-Stach, Stefan*, [**Periods and Nori motives**](http://dx.doi.org/10.1007/978-3-319-50926-6), Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 65. Cham: Springer (ISBN 978-3-319-50925-9/hbk; 978-3-319-50926-6/ebook). xxiii, 372 p. (2017). [ZBL1369.14001](https://zbmath.org/?q=an:1369.14001).


[2] *Deligne, Pierre; Milne, J.S.*, Tannakian categories, Hodge cycles, motives, and Shimura varieties, Lect. Notes Math. 900, 101-228 (1982). [ZBL0477.14004](https://zbmath.org/?q=an:0477.14004).


