proof of theorem about cyclic subspaces
We first prove the case . The inclusion is clear, since the right side is a -invariant subspace that contains .
Since , there exist polynomials and such that
Now is the projection from to :
(by assumption that is the annihilator polynomial of ) and
(by choice of and ), so
For the last claim, we note that the annihilator polynomial of is the least common multiple of and (that is a multiple of follows from the fact that must annihilate , and the set of polynomials that annihilate is the ideal generated by ). Since and are coprime, the lcm is just their product.
That concludes the proof for . If is arbitrary, we can simply apply the case inductively. We only have to check that the coprimality condition is preserved under applying the case to . But it is well-known that if (in or in any principal ideal domain) are pairwise coprime, then and are also coprime.
|Title||proof of theorem about cyclic subspaces|
|Date of creation||2013-03-22 17:32:53|
|Last modified on||2013-03-22 17:32:53|
|Last modified by||FunctorSalad (18100)|