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The development of the modern field of algebra has been deeply influenced by research in number theory.  An example of a problem from number theory that has provided far-reaching motivation for work in pure algebra has been the question of unique factorization.

The development of the modern field of algebra has been deeply influenced by research in number theory.  An example of a problem from number theory that has provided far-reaching motivation for work in pure algebra has been the question of unique factorization.

It's a well-known fact, the [[fundamental theorem of arithmetic]], that every integer can be factored in a unique way as the product of prime numbers.  It's natural to wonder whether this unique factorization holds in other simple rings, for example, the set of complex numbers whose real and complex parts are both integers (for example, <math>3+7i</math>).  These are the so-called Gaussian integers, and there is a good notion of "prime", and that every Gaussian integer can be uniquely factored into primes!  For example, 7+0i is a Gaussian prime, while 5+0i is not: <math>5+0i</math> factors as <math>(2+i)(2-i)<math>, and both <math>2-i</math> and <math>2+i</math> are prime.  In general, it turns out that  

It's a well-known fact, the [[fundamental theorem of arithmetic]], that every integer can be factored in a unique way as the product of prime numbers.  It's natural to wonder whether this unique factorization holds in other simple rings, for example, the set of complex numbers whose real and complex parts are both integers (for example, <math>3+7i</math>).  These are the so-called Gaussian integers, and there is a good notion of "prime", and that every Gaussian integer can be uniquely factored into primes!  For example, 7+0i is a Gaussian prime, while 5+0i is not: <math>5+0i</math> factors as <math>(2+i)(2-i)</math>, and both <math>2-i</math> and <math>2+i</math> are prime.  In general, it turns out that  

a Gaussian integer <math>a+bi</math> is prime if either one of <math>a</math> or <math>b</math> is zero, and the other is a prime congruent to three mod 4, or if <math>a</math> and <math>b</math> are nonzero, and <math>a^2+b^2</math> is a prime (in the usual sense).

a Gaussian integer <math>a+bi</math> is prime if either one of <math>a</math> or <math>b</math> is zero, and the other is a prime congruent to three mod 4, or if <math>a</math> and <math>b</math> are nonzero, and <math>a^2+b^2</math> is a prime (in the usual sense).