> R<x> := PolynomialRing(RationalField()); > K<a> := NumberField(x^3-2); > O := Order([2*a]); > O; Transformation of Order over Equation Order with defining polynomial x^3 - 2 over ZZ Transformation Matrix: [1 0 0] [0 2 0] [0 0 4] > OK := MaximalOrder(K); > Index(OK,O); 8 > Discriminant(O); -6912 > Discriminant(OK); -108 > 6912/108; 64 // perfect square... > R<x> := PolynomialRing(RationalField()); > K<a> := NumberField(x^2-7); > K<a> := NumberField(x^2-5); > Discriminant(K); 20 // ????????? Yuck! > OK := MaximalOrder(K); > Discriminant(OK); 5 // better > Discriminant(NumberField(x^2-20)); 80 > I := 7*OK; > I; Principal Ideal of OK Generator: [7, 0] > J := (OK!a)*OK; // the ! computes the natural image of a in OK > J; Principal Ideal of OK Generator: [-1, 2] > I*J; Principal Ideal of OK Generator: [-7, 14] > J*I; Principal Ideal of OK Generator: [-7, 14] > I+J; Principal Ideal of OK Generator: [1, 0] > > Factorization(I); [ <Principal Prime Ideal of OK Generator: [7, 0], 1> ] > Factorization(3*OK); [ <Principal Prime Ideal of OK Generator: [3, 0], 1> ] > Factorization(5*OK); [ <Prime Ideal of OK Two element generators: [5, 0] [4, 2], 2> ] > Factorization(11*OK); [ <Prime Ideal of OK Two element generators: [11, 0] [14, 2], 1>, <Prime Ideal of OK Two element generators: [11, 0] [17, 2], 1> ]We can even work with fractional ideals in .

> K<a> := NumberField(x^2-5); > OK := MaximalOrder(K); > I := 7*OK; > J := (OK!a)*OK; > M := I/J; > M; Fractional Principal Ideal of OK Generator: -7/5*OK.1 + 14/5*OK.2 > Factorization(M); [ <Prime Ideal of OK Two element generators: [5, 0] [4, 2], -1>, <Principal Prime Ideal of OK Generator: [7, 0], 1> ]

In the next chapter, we will learn about discriminants and an algorithm for ``factoring primes'', that is writing an ideal as a product of prime ideals of .

William Stein 2004-05-06