**1. Greatest common divisors. **

Let’s conclude the discussion from last lecture.

If is a field and are nonzero, then we can find polynomials such that is a gcd of and

To see this, consider and for some polynomials we have

We see that because both and are nonzero linear combinations of and so their degrees are in Each element of is a natural number because only for By the well-ordering principle, there is a least element of

Let be this least degree, and let have degree

First, if and then so

Second, by the division algorithm, we can write for some polynomials with Then is a linear combination of Since and is the smallest number in it follows that i.e., This is to say that so Similarly,

It follows that is a greatest common divisor of

Since any other greatest common divisor of is for some unit it follows that any gcd of and is a linear combination of and

Notice that this argument is very similar to the proof of the same result for