Polynomial ring/Field/One variable/Euclidean division/Fact/Proof

From Wikiversity
Jump to navigation Jump to search

We prove the statement about the existence by induction over the degree of . If the degree of is larger than the degree of , then and is a solution.

Suppose that . By the remark just made also holds, so is a constant polynomial, and therefore (since and is a field) and is a solution.

So suppose now that and that the statement for smaller degrees is already proven. We write and with . Then setting we have the relation

The degree of this polynomial is smaller than and we can apply the induction hypothesis to it. That means there exist and such that

From this we get altogether

so that and is a solution.

To prove uniqueness, let , both fulfilling the stated conditions. Then . Since the degree of the difference is smaller than , this implies and so .