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Could some one help me with understanding the notion of restriction of scalars? For instance in the case of algebraic group, for instance, $GL_N$, what does restriction of scalars from $K$ to $\mathbb{Q}$ mean, where $K/\mathbb{Q}$ is a finite field extension over the rational numbers?

How to embed $GL_n(K)$ inside $GL_N(\mathbb{Q})$ where $N= [K:Q] \times n$ concretely?

Thank you

Vanya
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  • Do you mean Weil restriction? – Qiaochu Yuan Mar 10 '16 at 18:30
  • These notes, which come up immediately if you google, seem to explain it very nicely: http://math.uga.edu/~pete/SC5-AlgebraicGroups.pdf – David Loeffler Mar 10 '16 at 23:00
  • Yeah, I mean Weil restriction. I wanted to see how the domain group sits in the target group, concretly. In one notes I searched for, the image of the source group is characterized as the set of matrices in the target group that commutes with the scalar multiplication map $\theta : K \to End_{\mathbb{Q}}(V_0)$ where $V_0$ is the vector space corresponding to $GL_n$. – Vanya Mar 11 '16 at 04:24
  • @David Loeffler, thank you very much. I will look into the notes – Vanya Mar 11 '16 at 04:25

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