Suppose the vector space is the direct sum of cyclic -modules whose annihilators are , , , and . Determine the invariant factors and elementary divisors of .

We have . If is irreducible over , then by the Chinese Remainder Theorem for modules (see here and here), we have .

The elementary divisors of are thus , , , , , , , and .

The invariant factors of are , , and .

If is reducible over , then is has a root , and indeed . Now by the Chinese Remainder Theorem for modules we have .

So the elementary divisors of are , , , , , , , , , and .

The invariant factors of are , , and . (Which are the same as if did not contain . This is to be expected in light of Corollary 18 on page 477 of D&F.)

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## Comments

You need to consider characteristic 2, then $( x+1)$ factors to $(x+i)^2$

All that matters here is how the polynomials factor over .