integral basis

A vector space has a basis: a small list of vectors from which every element is built by scalar combination. The ring of integers of a number field deserves the same kind of finite, rigid coordinate system — but with whole-number coefficients only, not arbitrary scalars. An integral basis is exactly such a coordinate system, letting us write every algebraic integer in the field with a unique tuple of ordinary integers.

Let K be a number field of degree n. An integral basis is a set b_1, ..., b_n of elements of O_K such that every element of O_K is uniquely a Z-linear combination m_1 b_1 + ... + m_n b_n with the m_i in Z. Such a basis exists because O_K is a free Z-module of rank n; equivalently the b_i are simultaneously a Q-basis of K and generate O_K over Z.

Different integral bases are related by matrices in GL(n, Z), so the determinant of the trace form det([Tr(b_i b_j)]) is independent of the choice — it is the discriminant of K. Finding an integral basis can be subtle: it is not always {1, a, a^2, ..., a^{n-1}} for a generator a, precisely when O_K is strictly larger than Z[a].