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[parent] minimality of integral basis (Theorem)

The discriminant $ \Delta := \Delta(\alpha_1,\,\alpha_2,\,\ldots,\,\alpha_s)$ of the set $ \{\alpha_1,\,\alpha_2,\,\ldots,\,\alpha_s\}$ of integers of an algebraic number field $ K$ is a rational integer. If this set is an integral basis of $ K$, then $ \vert\Delta\vert$ has the least possible (positive integer) value in the field $ K$, and conversely. The value $ d = \Delta$ is equal for all integral bases of $ K$, and it is called the discriminant or fundamental number of the field.



"minimality of integral basis" is owned by Mathprof. [ owner history (1) ]
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See Also: canonical basis

Also defines:  fundamental number, discriminant of field

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Cross-references: integral bases, field, positive, integral basis, algebraic number field, integers, discriminant
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This is version 6 of minimality of integral basis, born on 2005-06-17, modified 2008-02-21.
Object id is 7164, canonical name is MinimalityOfIntegralBasis.
Accessed 1869 times total.

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AMS MSC: 11R04 (Number theory :: Algebraic number theory: global fields :: Algebraic numbers; rings of algebraic integers)
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