Schemes with at most Five Elements

In this section, we shall see how some of the representation theoretic results of the previous sections can be applied successfully if S is small. 

Lemma 9.5.1 Let C be an algebraically closed field of characteristic 0. Then, 

We are assuming that Irr(CS) = 2. Thus, there exists an irreducible character x °f CS such that lcs x = Irr(CS ). 

Let us write Q to denote the smallest subfield of C, and let Z denote the smallest unitary subring of C. Then x( Ts) Q. On the other hand, by Lemma 9.2.5, x(as) is integral over Z. Thus, as C is assumed to have characteristic 0) x( Ts) G Z- It follows that 

This is an immediate consequence of Lemma 9.5.1 and Corollary 8.6.5. 

The second part of the following theorem is due to Paul Erdos, Alfred Renyi, and Vera Sos cf. [9 Theorem 6]. Its third part is a result of Alan Hoffman 

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Schemes with at most Five Elements 205 We start with two results due to Donald Higman cf. [20 (4.2)]. 

In the predominantly AAS scheme, phosphorus and titanium are colorimetrically determined. The results obtained on five replicates of the solution by each method are given in Table III and are compared with the values obtained for this standard at the Indiana Geological Survey.07) The ICPES and AAS results are in very good agreement with each other and with the literature values. The precision and the accuracy of the measurements are 5% for most elements analyzed. 

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