Weil restriction

In mathematics, restriction of scalars (also known as "Weil restriction") is a functor which, for any finite extension of fields L/k and any algebraic variety X over L, produces another variety Res_L/kX, defined over k. It is useful for reducing questions about varieties over large fields to questions about more complicated varieties over smaller fields.

Let L/k be a finite extension of fields, and X a variety defined over L. The functor ${\displaystyle \mathrm {Res} _{L/k}X}$ from k-schemes^op to sets is defined by

(In particular, the k-rational points of ${\displaystyle \mathrm {Res} _{L/k}X}$ are the L-rational points of X.) The variety that represents this functor is called the restriction of scalars, and is unique up to unique isomorphism if it exists.

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