proof of theorem about cyclic subspaces

We first prove the case r=2. The inclusion is clear, since the right side is a T-invariant subspace that contains v1+v2.

For the other inclusion, it is sufficient to show that v1,v2Z(v1+v2,T). The idea is that the action of T on v1+v2 can ”isolate” the two summands if their annihilator polynomials are coprimeMathworldPlanetmathPlanetmath. Let’s write mi for mvi.

Since (m1,m2)=1, there exist polynomialsMathworldPlanetmathPlanetmath p and q such that

pm1+qm2=1 (1)

this is Bézout’s lemma (or the Euclidean algorithmMathworldPlanetmath, or the fact that k[X] is a principal ideal domainMathworldPlanetmath).

Now pm1(T) is the projectionPlanetmathPlanetmathPlanetmath from Z(v1,T)Z(v2,T) to Z(v2,T):

(pm1)(T)v1=p(T)m1(T)v1=p(T)0=0 (2)

(by assumption that m1 is the annihilator polynomial of v1) and

(pm1)(T)=1-(qm2)(T) (3)

(by choice of p and q), so

(pm1)(T)v2=v2-q(T)m2(T)v2=v2-q(T)0=v2 (4)

Any subspacePlanetmathPlanetmath that is invariant under T is also invariant under polynomials of T. Therefore, the preceding equations show that v2=(pm1)(T)(v1+v2)Z(v1+v2,T). By symmetry, we also get that v1Z(v1+v2,T).

For the last claim, we note that the annihilator polynomial m of Z(v1,T)Z(v2,T) is the least common multipleMathworldPlanetmath of m1 and m2 (that m is a multiple of m1 follows from the fact that m must annihilate v1, and the set of polynomials that annihilate v1 is the ideal generated by m1). Since m1 and m2 are coprime, the lcm is just their product.

That concludes the proof for r=2. If r is arbitrary, we can simply apply the r=2 case inductively. We only have to check that the coprimality condition is preserved under applying the r=2 case to i=1,2. But it is well-known that if p,q,r (in k[X] or in any principal ideal domain) are pairwise coprime, then pq and r are also coprime.

Title proof of theorem about cyclic subspaces
Canonical name ProofOfTheoremAboutCyclicSubspaces
Date of creation 2013-03-22 17:32:53
Last modified on 2013-03-22 17:32:53
Owner FunctorSalad (18100)
Last modified by FunctorSalad (18100)
Numerical id 7
Author FunctorSalad (18100)
Entry type Proof
Classification msc 15A04