Scheme Rings over the Field of Complex Numbers

Let us denote by Z the smallest unitary subring of C. The C-module CS is finitely generated. Thus, is zero of a monic polynomial over Z. Thus, each characteristic root of Ts is integral over Z. Thus, as x(fs) is a sum of characteristic roots of ag, the claim follows from Theorem 8.2.4. 

In this section, the letter C stands for the field of complex numbers. The scheme ring over the field of complex numbers provides us with an additional tool to work with, the norm function of C. 

Since c is a characteristic root of (Tg, there exists an element m in CX such that m and mag = cm. 

Since m G CX, there exists, for each element x in X, an element Cj, in C such that 

Let us now pick an element z in X such that, for each of the finitely many elements x in X, cx cz. Then, by Lemma 1.1.2(i), 

Lemma 9.2.5 Let x be an irreducible character of CS, and let s be an element in S. Then x as) integral over the smallest unitary subring of C. 

---

Scheme Rings over the Field of Complex Numbers 195... 

---