What does it mean for something to be reducible over Q or Z?

[sick grove]

What does it mean for something to be reducible over Q or Z?

[cinder ore]

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[sturdy hinge]

In what context?

[onyx mango]

it usually refers to a polynomial, where a polynomial P, isn’t reducible. Like $x^2+1$ that polynomial is irreducible over the rationals and integers. a polynomial
$f(x)∈Q[x]$ is reducible if f can be expressed as a product of two functions (factors if you may) Such that g(x) and h(x) are non constant polynomials of Q and for Z you basically can reduce the polynomial P to integer factors.

[brisk crypt]

ビジヨン
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ビジヨン

[onyx mango]

the cubic $x^3-6x-9$ Is reducible over both Q and Z as one of its roots it 3 thus giving $(x-3)(x^2+3x+3)$ now the quadratic, is it reducible over Q and Z?

[brisk crypt]

ビジヨン

[solid kite]

A polynomial $A(x)$ is reducible over some field $\mathbb F$ if it can be expressed as $B(x)C(x)$ where $B, C$ are also polynomial over that field and $\deg B, \deg C <\deg A$.

[brisk crypt]

Coffey

[solid kite]

For example

$X^2+1$ is irreducible over $\bZ, \bQ, \bR$ but it's reducible over $\bC$ as, $$X^2+1=(X-i)(X+i)$$

[brisk crypt]

Coffey

[solid kite]

also check out "Gauss' lemma on reduciblity"

[sick grove]

+close