Loterre: Mathématiques: ring of integers

algebra > commutative algebra > commutative ring > ring of integers

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ring of integers

In mathematics, the ring of integers of an algebraic number field ${\\displaystyle K}$ is the ring of all algebraic integers contained in ${\\displaystyle K}$ . An algebraic integer is a root of a monic polynomial with integer coefficients: ${\\displaystyle x^{n}+c_{n-1}x^{n-1}+\\cdots +c_{0}}$ . This ring is often denoted by ${\\displaystyle O_{K}}$ or ${\\displaystyle {\\mathcal {O}}_{K}}$ . Since any integer belongs to ${\\displaystyle K}$ and is an integral element of ${\\displaystyle K}$ , the ring ${\\displaystyle \\mathbb {Z} }$ is always a subring of ${\\displaystyle O_{K}}$ .
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Ring_of_integers)