Let be the set of polynomials with complex coefficients. For every and , there exist such that True or false?
True!
For every factor of , and every , we multiply by an appropriate number of factors of the form , so as to obtain a polynomial which can be written as a product of powers of polynomials of the form .
We write , with . We let be a primitive root of in . Now, we let . A quick check shows that the coefficients of lie in . Now forget that the are polynomials in ; the coefficients of , as a polynomial in , are symmetric polynomials in with coefficients in . But since , only the first and the last symmetric polynomial invariants do not vanish, and hence only powers of remain, i.e. ...
this problem has a nice result in linear algebra: for a field let be the field of rational functions in the indeterminate i.e.
let then is obviously a subfield of and therefore we can consider as a vector space over using what we just proved, it's easy to see that every element of
can be written uniquely as thus we get the result which is basically the reason that i created this thread!