By Atle Selberg
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Extra info for Atle Selberg Collected Papers
It is clear that Z can be embedded into ZŒi. However not every rational prime is also prime in ZŒi. The primes in ZŒi are called the Gaussian primes. For example we can show that both 1 C i and 1 i are Gaussian primes, that is primes in ZŒi. 1 i / D 2 so that the rational prime 2 is not a prime in ZŒi. Using the multiplicativity of the Euclidean norm in ZŒi we can describe all the units and primes in ZŒi. 9. (2) Suppose (1) The only units in ZŒi are ˙1; ˙i. is a Gaussian prime. Then is either: (a) a positive rational prime p Á 3 mod 4 or an associate of such a rational prime.
4 The Existence of Maximal Ideals In this section we prove that in any ring R with an identity there do exist maximal ideals. Further given an ideal I ¤ R then there exists a maximal ideal I0 such that I I0 . To prove this we need three important equivalent results from logic and set theory. First recall that a partial order Ä on a set S is a reﬂexive, transitive relation on S. That is a Ä a for all a 2 S and if a Ä b; b Ä c then a Ä c. This is a “partial” order since there may exist elements a 2 S where neither a Ä b nor b Ä a.
4 we get c D ˙1. Hence a D ˙3. This is a contradiction, so 3 does not divide a. An identical argument shows that 3 does not divide b. Therefore 3 is not a prime element in R. We now examine the relationship between prime elements and irreducibles. 6. Let R be an integral domain. Then (1) Each prime element of R is irreducible. (2) p 2 R is a prime element if and only if p ¤ 0 and hpi D pR is a prime ideal. (3) p 2 R is irreducible if and only if p ¤ 0 and hpi D pR is maximal in the set of all principal ideals of R which are not equal to R.