By Peter Barlow

Barlow P. An straightforward research of the idea of numbers (Cornell collage Library, 1811)(ISBN 1429700467)

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Equivalently, every algebraic number can be expressed as the quotient of an algebraic integer by an element of Z. 11 Show that √ 1+ 5 2 √ 1+ √ 3 2 is an algebraic integer. is an algebraic integer. 12 Let a be an integer. Show that α = (1 + a 1/3 + a 2/3 )/3 is a root of X3 − X2 + (1 − a)2 1−a X− = 0. ] Deduce that if a ≡ 1 (mod 9), then α is an algebraic integer. 6 The Ring of All Algebraic Integers We will want to study factorisation and so on in number fields. This will require a definition of integers and primes in these fields.

9 that the embedding ιk extends to an embedding K −→ C in rα ways; by definition of an extension of embeddings, each extension maps α to αk . We can do this for each of the dα embeddings ιk , extending each in rα ways. We thus obtain dα rα = n embeddings from K to C. 7. We therefore conclude that all of the embeddings σi : K −→ C have been obtained, and we have seen that α is taken to each of its conjugates {α1 , . . , αdα } with multiplicity rα . 12 Suppose α in K has minimal polynomial g of degree dα , and that rα = n/dα .

Then A and B will be mn × mn-matrices, α + β, α − β and αβ are easily seen to be eigenvalues of A + B, A − B and AB respectively (with v as eigenvector), and the characteristic polynomials of A + B, A − B and AB have degree mn. Further, notice that if α and β are both algebraic integers, then the matrices A and B have entries in Z, and so the entries of A + B, A − B and AB are all also in Z. Therefore the characteristic polynomials of these three matrices are all integral, and are monic by definition, so this gives another proof that the eigenvalues α + β, α − β and αβ are all algebraic integers.