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001. About how many? 13. Show that 17 is the norm of an element of Z[ J2]. 14. Show that if n is the norm of an element of Z[J2], then so is -n. 15. Working in Z[v'3] instead of Z[J2], formulate and prove an identity similar to the two-squares identity for numbers of the form a 2 - 3b2 • 16. Can you give a reasonable definition of the norm of an element of Z1o? • If a and b are elements of any ring R , we say that a divides b if b = ca for some c in R. The rational numbers were constructed so that we could divide by any nonzero integer: because 2 does not divide 1 in the integers, the number 1/ 2 was created so that 2 would divide 1 somewhere.

Integers that divide everything. One and minus one. 3. Primes. Integers which are not units, but cannot be written as a product unless one factor is a unit. In this chapter we will examine the units in the various rings we have introduced and in the next chapter do the same for primes. 1 Elementary properties • The number 1 has the property that 1 · m == m · 1 == m for m in the ring, and it is the only number with this property. An element u of a ring is a unit if uv = vu = 1 for some v in the ring; the element v is called the 1 inverse of u, and is often denoted u- .

The first n terms are 2 3 u, u 'u ' ... 'u n = 1 and, because n is the order of u, only un is equal to 1. But un+l = unu = u, and un+2 = unu 2 = u 2 , and so on. That is, the sequence is periodicthe displayed block of n units simply repeats over and over again in the sequence u, u 2 , u 3 , . . • Every nth term is 1, and those are the only terms which are 1. Corollary 20. If u is a unit of order n, and d is the greatest common divisor of n and k, then the order of uk is n/ d. Proof. From the theorem, (uk)m = 1 if and only if n divides km.