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Arithmetic in Extensions of Q

What makes work with rational numbers and integers comfortable are the essential properties they have, especially the unique factorization property (the Main Theorem of Arithmetic). However, the might of the arithmetic in Q is bounded. Thus, some polynomials, although they have zeros, cannot be factorized into polynomials with rational coefficients. | The IMO Compendium jroup www.imo.orfl.yu 2007 The Author s and The IMO Compendium Group Ofympiaif Iraininff Materials www.imo compendium com Arithmetic in Extensions of Q Dusan DjukiC Contents 1 General Properties . 1 2 Arithmetic in the Gaussian Integers Z i . 4 3 Arithmetic in the ring Z w . 5 4 Arithmetic in other quadratic rings. 6 1 General Properties What makes work with rational numbers and integers comfortable are the essential properties they have especially the unique factorization property the Main Theorem of Arithmetic . However the might of the arithmetic in Q is bounded. Thus some polynomials although they have zeros cannot be factorized into polynomials with rational coefficients. Nevertheless such polynomials can always be factorized in a wider field. For instance the polynomial x2 1 is irreducible over Z or Q but over the ring of the so called Gaussian integers Z i a bi a b G Z it can be factorized as x i x - i . Sometimes the wider field retains many properties of the rational numbers. In particular it will turn out that the Gaussian integers are a unique factorization domain just like the rational integers Z. We shall first discuss some basics of higher algebra. Definition 1. A number a G C is algebraic if there is a polynomial p x anxn an-1xn-1 a0 with integer coefficients such that p a 0. If an 1 then a is an algebraic integer. Further p x is the minimal polynomial of a if it is irreducible over Z x i.e. it cannot be written as a product of nonconstant polynomials with integer coefficients . Example 1. The number i is an algebraic integer as it is a root of the polynomial x2 1 which is also its minimal polynomial. Number y 2 yC is also an algebraic integer with the minimal polynomial x4 10x2 1 verify . Example 2. The minimal polynomial of a rational number q a b a G Z b G N a b 1 is bx a. By the definition q is an algebraic integer if and only if b 1 i.e. if and only if q is an integer. Definition 2. Let a be an algebraic integer andp x xn .