In mathematics, the discriminant of an algebraic number field is a numerical invariant that, loosely speaking, measures the size of the ( ring of integers of the) algebraic number field. More specifically, it is related to the volume of the fundamental domain of the ring of integers, and it regulates which primes are ramified.
The discriminant is one of the most basic invariants of a number field, and occurs in several important analytic formulas such as the functional equation of the Dedekind zeta function of K, and the analytic class number formula for K. An old theorem of Hermite's states that there are only finitely many number fields of bounded discriminant, however determining this quantity is still an open problem, and the subject of current research. [1]
Let K be an algebraic number field, and let OK be its ring of integers. Let b1, ..., bn be an integral basis of OK (i.e. a basis as a Z-module), and let {σ1, ..., σn} be the set of embeddings of K into the complex numbers (i.e. ring homomorphisms K→C). The discriminant of K is the square of the determinant of the n by n matrix whose (i,j)-entry is σi(bj). Symbolically,
Equivalently, the
trace from K to Q can be used. Specifically, define the
trace form to be the matrix whose (i,j)-entry is
TrK/Q(bibj). Then the discriminant of K is the determinant of this matrix.
The discriminant defined above is sometimes referred to as the absolute discriminant of K to distinguish it from the relative discriminant ΔK/L of an extension of number fields K/L, which is an ideal in OL. When L = Q, the relative discriminant ΔK/Q is the principal ideal generated by the absolute discriminant ΔK.
Hermite's theorem states that there are only finitely many algebraic number fields of bounded discriminant; the question of the exact number, for a given bound, has proved to be a difficult one. It has generally been attacked by fixing the degree of the number field (and also the Galois group of the Galois closure).
Similar conjectures exist for relative discriminants as well.
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help)In mathematics, the discriminant of an algebraic number field is a numerical invariant that, loosely speaking, measures the size of the ( ring of integers of the) algebraic number field. More specifically, it is related to the volume of the fundamental domain of the ring of integers, and it regulates which primes are ramified.
The discriminant is one of the most basic invariants of a number field, and occurs in several important analytic formulas such as the functional equation of the Dedekind zeta function of K, and the analytic class number formula for K. An old theorem of Hermite's states that there are only finitely many number fields of bounded discriminant, however determining this quantity is still an open problem, and the subject of current research. [1]
Let K be an algebraic number field, and let OK be its ring of integers. Let b1, ..., bn be an integral basis of OK (i.e. a basis as a Z-module), and let {σ1, ..., σn} be the set of embeddings of K into the complex numbers (i.e. ring homomorphisms K→C). The discriminant of K is the square of the determinant of the n by n matrix whose (i,j)-entry is σi(bj). Symbolically,
Equivalently, the
trace from K to Q can be used. Specifically, define the
trace form to be the matrix whose (i,j)-entry is
TrK/Q(bibj). Then the discriminant of K is the determinant of this matrix.
The discriminant defined above is sometimes referred to as the absolute discriminant of K to distinguish it from the relative discriminant ΔK/L of an extension of number fields K/L, which is an ideal in OL. When L = Q, the relative discriminant ΔK/Q is the principal ideal generated by the absolute discriminant ΔK.
Hermite's theorem states that there are only finitely many algebraic number fields of bounded discriminant; the question of the exact number, for a given bound, has proved to be a difficult one. It has generally been attacked by fixing the degree of the number field (and also the Galois group of the Galois closure).
Similar conjectures exist for relative discriminants as well.
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