I came up with this little problem last night. It’s not very difficult to prove but still fun (I think). Here it is : let be a commutative ring, and let . Suppose that, for any polynomials , we have . Then .

I’ll post my solution in a couple of days to see if anyone can come up with an alternative solution in the meantime. :)

Let be relatively prime integers greater than the degree of . Then the numbers with not greater than degree of are all different (exercise!), and so the equality:

implies that for all :D

Perfect solution! It’s exactly the same solution that I found. I wonder if there are any other simple solutions. This is probably as simple as it gets, though.

Nice problem! :) Here is my solution:

Suppose that , then .

Let be relatively prime integers greater than the degree of . Then the numbers with not greater than degree of are all different (exercise!), and so the equality:

implies that for all :D

Perfect solution! It’s exactly the same solution that I found. I wonder if there are any other simple solutions. This is probably as simple as it gets, though.

Congratulations! :D