Let be a polynomial with real coefficients. Show that there exists a nonzero polynomial with real coefficients such that has terms that are all of a degree divisible by .
Here's one way to approach this problem. To get an idea of what is involved, start with a simpler problem.
Let be a polynomial with real coefficients. Show that there exists a nonzero polynomial with real coefficients such that has terms that are all of a degree divisible by 2.
A bit of thought shows that this is not too hard. Split P(x) into terms of even degree and terms of odd degree. Then we can write , where both A(x) and B(x) have all terms of even degree. If then , which has all terms of even degree, as required. For convenience later on, define .
Now try something a bit harder.
Let be a polynomial with real coefficients. Show that there exists a nonzero polynomial with real coefficients such that has terms that are all of a degree divisible by 3.
Let , a complex cube root of unity. Check that if is a real quadratic polynomial then has terms that are all of a degree divisible by 3. Also, the second and third of those three factors are complex conjugates of each other, so their product is a real polynomial.
For a general polynomial P(x), we can write , where each of A(x), B(x) and C(x) has terms that are all of a degree divisible by 3. Let . Then is a multiple of P(x) whose terms are all of a degree divisible by 3.
The next part of the project is to do the same thing with 3 replaced by 5. You obviously have to start with a fifth complex root of unity, , and the argument gets a bit tedious, so I'll skip it. The end result is a real polynomial Q(x) (a product of four complex factors) such that is a multiple of P(x) whose terms are all of a degree divisible by 5.
Now you can start to stitch things together. Notice that is a multiple of P(x) whose terms are all of a degree divisible by 10. Then inductively define . So finally, is a multiple of P(x) whose terms are all of a degree divisible by .