PlanetMath: ring without irreducibles

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ring without irreducibles

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An integral domain may not contain any irreducible elements. One such example is the ring of all algebraic integers. Any non-zero non-unit $\vartheta$ of this ring satisfies an equation

$\displaystyle x^n\!+\!a_1x^{n-1}\!+\!\cdots\!+\!a_{n-1}x\!+\!a_n = 0$

, since it is an algebraic integer; moreover, we can assume that

N $(\vartheta) \neq \pm 1$ (see norm and trace of algebraic number: theorem 2). The element $\vartheta$ has the decomposition

$\displaystyle \vartheta = \sqrt{\vartheta}\!\cdot\!\sqrt{\vartheta}.$

Here, $\sqrt{\vartheta}$ belongs to the ring because it satisfies the equation

$\displaystyle x^{2n}\!+\!a_1x^{2n-2}\!+\!\cdots\!+\!a_{n-1}x^2\!+\!a_n = 0,$

and it is no unit. Thus the element $\vartheta$ is not irreducible.

"ring without irreducibles" is owned by pahio.

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See Also: algebraic integer

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13G05 (Commutative rings and algebras :: Integral domains)

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