Discriminant of An Algebraic Number Field - Relative Discriminant

The discriminant defined above is sometimes referred to as the absolute discriminant of K to distinguish it from the relative discriminant Δ_K/L of an extension of number fields K/L, which is an ideal in O_L. The relative discriminant is defined in a fashion similar to the absolute discriminant, but must take into account that ideals in O_L may not be principal and that there may not be an integral basis of K/L. Let {σ₁, ..., σ_n} be the set of embeddings of K into C which are the identity on L. If b₁, ..., b_n is any basis of K over L, let d(b₁, ..., b_n) be the square of the determinant of the n by n matrix whose (i,j)-entry is σ_i(b_j). Then, the relative discriminant of K/L is the ideal generated by the d(b₁, ..., b_n) as {b₁, ..., b_n} varies over all bases of K/L with the property that b_i ∈ O_K for all i. Alternatively, the relative discriminant of K/L is the norm of the different of K/L. When L = Q, the relative discriminant Δ_K/Q is the principal ideal of Z generated by the absolute discriminant Δ_K . In a tower of fields K/L/F the relative discriminants are related by

where denotes relative norm.