Discriminants and integral bases

For a Q-basis $b_1,\ldots,b_n$ of a number field K, its discriminant is the determinant of the matrix with entries $\operatorname{Tr}_{K/\mathbb Q}(b_ib_j)$. Separability makes this determinant nonzero. Equivalently it is the square of the determinant of the embeddings matrix. The sign can be negative when complex embeddings occur, so this is not simply a positive Euclidean squared volume. A change of basis by matrix A multiplies the discriminant by det(A) squared.

When the basis is integral, every trace entry is an integer. If it is a Z-basis of the full ring of integers, its discriminant is the field discriminant d_K, independent of the chosen integral basis because unimodular changes have determinant plus or minus one. An integral collection that spans only a proper order gives that order's discriminant instead. Calling such a collection an integral basis merely because each member is integral confuses an elementwise property with a lattice-basis property.

If an order O has finite index I in $\mathcal O_K$, its lattice basis change has determinant of absolute value I. Hence disc(O)=I²d_K. For an integral primitive element θ, the discriminant of its monic minimal polynomial is the discriminant of the power basis, therefore I²d_K with $I=[\mathcal O_K:\mathbb Z[\theta]]$. This formula supplies a constraint on possible missing integral elements: the squared index must divide the power-order discriminant.

It does not say that every square divisor actually occurs as an index. An index calculation needs the integer ring or an argument bounding and excluding alternatives. Conversely, a squarefree power-basis discriminant forces index one because no nontrivial index square can divide it. Replacing an integral basis by a rationally scaled one can produce a noninteger discriminant, so divisibility statements must specify that the compared lattices are integral full-rank orders. The field itself has not changed when a basis or order changes.

For a number field over Q, a rational prime ramifies in its full integer ring exactly when it divides the field discriminant. Divisibility of an arbitrary defining polynomial's discriminant is not the same test when the prime divides the power-order index. Such an index prime can make polynomial reduction appear repeated even though the field extension is unramified there. Separating these possibilities is one reason integral-basis calculations precede reliable prime-splitting computations.

The discriminant also enters global lattice estimates and the different ideal, so normalization mistakes propagate beyond ramification. In a quadratic field the explicit squarefree classification gives d_K=d when d is one modulo four, and d_K=4d otherwise. The determinant calculation should reproduce that formula rather than be replaced by memorization alone. Checking both the trace matrix and the index relation gives two independent ways to diagnose a missing factor of four or an incorrect choice of quadratic generator.