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integral basis (Definition)

Let $ K$ be a number field. A set of algebraic integers $ \{\alpha_1,\ldots,\alpha_s\}$ is said to be an integral basis for $ K$ if every $ \gamma$ in $ \mathcal{O}_K$ can be represented uniquely as an integer linear combination of $ \{\alpha_1,\ldots,\alpha_s\}$ (i.e. one can write $ \gamma = m_1 \alpha_1 + \cdots + m_s \alpha_s$ with $ m_1,\ldots,m_s$ (rational) integers).

If $ I$ is an ideal of $ \mathcal{O}_K$, then $ \{\alpha_1,\ldots,\alpha_s\} \in I$ is said to be an integral basis for $ I$ if every element of $ I$ can be represented uniquely as an integer linear combination of $ \{\alpha_1,\ldots,\alpha_s\}$.

(In the above, $ \mathcal{O}_K$ denotes the ring of algebraic integers of $ K$.)

An integral basis for $ K$ over $ \mathbb{Q}$ is a basis for $ K$ over $ \mathbb{Q}$.



"integral basis" is owned by rspuzio. [ full author list (2) | owner history (1) ]
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See Also: algebraic integer, integral, basis, discriminant, condition for power basis, basis of ideal in algebraic number field

Other names:  minimal basis, minimal bases
Also defines:  integral bases
Keywords:  integral basis

Attachments:
minimality of integral basis (Theorem) by Mathprof
canonical basis (Theorem) by pahio
power basis over $\mathbb{Z}$ (Definition) by Wkbj79
integral basis of quadratic field (Derivation) by pahio
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Cross-references: basis, ring, ideal, rational, linear combination, integer, algebraic integers, number field
There are 6 references to this entry.

This is version 9 of integral basis, born on 2002-04-19, modified 2006-04-21.
Object id is 2853, canonical name is IntegralBasis.
Accessed 5878 times total.

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