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# characterizations of integral

###### Theorem.

Let $R$ be a subring of a field $K$, $1\in R$ and let $\alpha$ be a non-zero element of $K$. The following conditions are equivalent:

Proof. Supposing the first condition means that an equation

$\alpha^{n}+a_{1}\alpha^{{n-1}}+\ldots+a_{{n-1}}\alpha+a_{n}=0,$ |

with $a_{j}$’s belonging to $R$, holds. Dividing both sides by $\alpha^{{n-1}}$ gives

$\alpha=-a_{1}-a_{2}\alpha^{{-1}}-\ldots-a_{n}\alpha^{{-n+1}}.$ |

One sees that $\alpha$ belongs to the ring $R[\alpha^{{-1}}]$ even being a unit of this (of course $\alpha^{{-1}}\in R[\alpha^{{-1}}]$). Therefore also the principal ideal $\alpha^{{-1}}R[\alpha^{{-1}}]$ of the ring $R[\alpha^{{-1}}]$ coincides with this ring. Conversely, the last circumstance implies that $\alpha$ is integral over $R$.

# References

- 1 Emil Artin: Theory of Algebraic Numbers. Lecture notes. Mathematisches Institut, Göttingen (1959).

Keywords:

integral over a ring

Synonym:

characterisations of integral

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Theorem

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Reference

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## Mathematics Subject Classification

12E99*no label found*13B21

*no label found*

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