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