7.4. A DIRECT SUM DECOMPOSITION 157

vector space over F. As usual, for α ∈ F [x]≡ D,αm≡ α (L)m. For p a monic irreduciblepolynomial,

Vp ≡{

m ∈V : pkm = 0 for some k ∈ N}

That is, eventually pkm = 0. It might be possible that k could change for different m ∈ V .Note that if p is invertible, then Vp = 0 because xpm = 0 if and only if m =

(x−1)p 0 = 0, so

nothing is lost from considering only irreducible non constant polynomials. It is obviousthat Vp is a subgroup of the module V and is itself a module. Indeed, if m∈Vp so that pkm=0 for some k, then 0 = α pkm = pkαm = 0 also and so αm ∈ V . If m, m̂ ∈ Vp, then lettingkm,km̂ be the exponents for m, m̂, let k ≥max(km,km̂) and pk (m+ m̂) = pkm+ pkm̂ = 0 sothe sum m+ m̂ is in Vp if m, m̂ are.

Proposition 7.4.1 Let p1, · · · , pn be monic irreducible nonconstant polynomials. Let V bea finite dimensional vector space. Then(

Vp1 + · · ·+Vp j−1 +Vp j+1 + · · ·+Vpn

)∩Vp j = 0

and so ∑i Vpi =⊕

i Vpi .

Proof: This follows from the observation that ∏i̸= j pkii and p

k jj are relatively prime. If

q is monic and divides the second, then it is of the form pm jj ,m j ≤ k j. If q divides the

first, then q is ∏i̸= j pmii ,mi ≤ ki. Thus ∏i ̸= j pmi

i = pm jj contradicting Theorem 1.13.9 about

uniqueness of factorization. Since the irreducible polynomials are distinct, we must have allm j,mi equal to 0 and q = 1 so these two, ∏i̸= j pki

i and pk jj are relatively prime as claimed. If

m ∈(

Vp1 + · · ·+Vp j−1 +Vp j+1 + · · ·+Vpn

)∩Vp j , then m = ∑i ̸= j mi and so there exist ki,k j

such that pkii mi = 0 and p

k jj m = 0. Since ∏i̸= j pki

i and pk jj are relatively prime, there exist

σ ,τ such that1 = σ ∏

i ̸= jpki

i + τ pk jj (*)

Then do both sides of ∗ to m.

m =

(σ ∏

i ̸= jpki

i

)(=m

∑i ̸= j

mi

)+ τ p

k jj m = 0

This yields m = 0 and verifies the conclusion of the proposition.It follows from Lemma 6.0.2 that if mi ∈ Vpi , and if ∑i mi = 0, then each mi = 0 so

∑i Vpi =⊕

i Vpi . ■

Lemma 7.4.2 Let V be a vector space over a field F and let L ∈L (V,V ) have minimumpolynomial α (x) = ∏

ni=1 pki

i (x) in which the pi are distinct irreducible monic polynomials.Let D≡ F [x] and for α ∈ D,α (m)≡ α (L)(m). Let

V̂pi ≡{

m ∈V : pkii m = 0

}⊆Vpi , (7.4)

(Note that here ki is fixed. ) Then

Dm⊆ V̂p1 ⊕·· ·⊕V̂pn ⊆Vp1 ⊕·· ·⊕Vpn (7.5)