Number fields and rings of integers

A number field K is a finite extension of $\mathbb Q$, with degree n. An element is an algebraic integer if it satisfies a monic polynomial in $\mathbb Z[x]$. The full set of such elements in K is $\mathcal O_K$, the integral closure of Z in K. Algebraic integers form a ring; this uses closure properties of integral extensions, not the claim that adding two roots of arbitrary equations simply adds their equations. Every rational number that is integral over Z is an ordinary integer.

Monicity is essential. Every algebraic number satisfies some nonzero integer polynomial after denominators are cleared, but that weaker property does not characterize integrality. For example a rational fraction can satisfy an integer polynomial with leading coefficient greater than one while failing every monic integer equation. An element's monic minimal polynomial over Q has integer coefficients exactly when the element is an algebraic integer. This criterion is useful for computations, provided the polynomial used really is the minimal one rather than an unrelated multiple.

The additive group of $\mathcal O_K$ is free of rank n. An integral basis is a Z-basis of that group and is also a Q-basis of K. The converse fails: a Q-basis made of integral elements may span only a proper finite-index sublattice of the full integer ring. Likewise, if an integral element θ generates K as a field, the power ring $\mathbb Z[\theta]$ is a full-rank order but need not equal $\mathcal O_K$. Not every number field has a power integral basis.

An order is a subring containing one whose additive group is a full-rank lattice in the number field. Its finite index in the maximal order measures a genuine arithmetic defect, not an error in the field degree. Passing from one order to another can change which ideals behave well under factorization and whether a polynomial reduction reads prime splitting correctly. It is therefore useful to distinguish field generation, integral generation, and additive lattice generation before attempting any modular or discriminant calculation.

For squarefree integer d different from zero and one, the ring of integers of $\mathbb Q(\sqrt d)$ is $\mathbb Z[(1+\sqrt d)/2]$ when d is congruent to one modulo four, and $\mathbb Z[\sqrt d]$ otherwise. This statement includes negative squarefree d. The half-integral basis arises because trace and norm can both be integral for those particular halves. It does not mean that dividing any integral element by two preserves integrality.

A derivation writes an arbitrary quadratic element in rational coordinates, uses its trace and norm to constrain denominators, and exploits squarefreeness. The resulting classification is stronger than checking a few convenient elements. In higher degree, finding the full integer ring can require more than detecting one missing half-integer. An explicit quadratic calculation is a good reference case for an algorithm: verify the monic polynomial, the additive basis relation, and the index separately rather than assuming that one successful check establishes all three.