The Discriminant of a Field and its Divisors

David Hilbert

Chapter 4 in The Theory of Algebraic Number Fields, 1998, pp 25-32 from Springer

Abstract: Abstract The discriminant d of a field k is defined by the equation $$d = \left| {\begin{array}{*{20}{c}} {{\omega _1}} \\ {{{\omega '}_1}} \\ {\omega _1^{(m - 1)}} \end{array}} \right.\begin{array}{*{20}{c}} \ldots \\ \ldots \\ { \ldots ...} \\ \ldots \end{array}{\left. {\begin{array}{*{20}{c}} {{\omega _m}} \\ {{{\omega '}_m}} \\ {\omega _m^{(m - 1)}} \end{array}} \right|^2}$$ where ω 1, ... , ω m form a basis for k; the discriminant d is a rational integer. For the development of field theory the investigation of the ideal factors of the discriminant is of central importance. We have the following fundamental theorem.

Date: 1998
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DOI: 10.1007/978-3-662-03545-0_4

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