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Primary decomposition over $Q$ with one linear and one irreducible quadratic factor

Example

On $V = Q \oplus Q^{2}$ , let $T$ act by multiplication by $2$ on the first summand and by

$C = (\begin{array}{cc} 0 & - 1 \\ 1 & 0 \end{array})$

on the second. Then $μ_{T} = (x - 2) (x^{2} + 1)$ , the primary components are the displayed summands, and their polynomial projections are

$E_{1} = \frac{T^{2} + I}{5}, E_{2} = \frac{4 I - T^{2}}{5} .$

Facts & Assumptions

Given: The rational endomorphism $T$ in the Example.

[L1]

Irreducible-power factors of a minimal polynomial give the direct sum of their kernels (Primary decomposition: the irreducible-power factors of $μ_{T}$ split $V$ into their invariant kernels).

[L2]

The associated primary projections are polynomial expressions in the endomorphism (Each projection in the primary decomposition is a polynomial in the endomorphism).

[L3]

The polynomial $x^{2} + 1$ is irreducible over $R$ , hence also over its subfield $Q$ ( $x^{2} + 1$ is irreducible over $R$ ).

Verification

technique · direct

1.1

The first block has minimal polynomial $x - 2$ , and $C^{2} = - I$ makes the second block's minimal polynomial $x^{2} + 1$ ; [L3] rules out a rational linear factor. The two factors are coprime, so the direct sum has minimal polynomial their product and [L1] identifies the two primary components.

L1L3algebra

2.1

The polynomial $e_{1} = (x^{2} + 1) / 5$ is $1$ at $x = 2$ and is $0$ modulo $x^{2} + 1$ ; $e_{2} = 1 - e_{1} = (4 - x^{2}) / 5$ has the opposite residues. Therefore [L2] gives the displayed projections.

L2algebra∎

Depends on

Used by

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Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 39 results over 8 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Anthony W. Knapp, Basic Algebra, 2nd ed., Ch. V, §5, Theorem 5.19 (standard reference, not scraped)

Primary decomposition over QQ with one linear and one irreducible quadratic factor