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In mathematics, a fundamental discriminant ''D'' is an integer invariant in the theory of integral binary quadratic forms. If is a quadratic form with integer coefficients, then is the discriminant of ''Q''(''x'', ''y''). Conversely, every integer ''D'' with is the discriminant of some binary quadratic form with integer coefficients. Thus, all such integers are referred to as ''discriminants'' in this theory. Every discriminant may be written as :''D'' = ''D''0''f'' 2 with ''D''0 a discriminant and ''f'' a positive integer. A discriminant ''D'' is called a fundamental discriminant if ''f'' = 1 in every such decomposition. Conversely, every discriminant ''D'' ≠ 0 can be written uniquely as ''D''0''f'' 2 where ''D''0 is a fundamental discriminant. Thus, fundamental discriminants play a similar role for discriminants as prime numbers do for all integers. There are explicit congruence conditions that give the set of fundamental discriminants. Specifically, ''D'' is a fundamental discriminant if, and only if, one of the following statements holds * ''D'' ≡ 1 (mod 4) and is square-free, * ''D'' = 4''m'', where ''m'' ≡ 2 or 3 (mod 4) and ''m'' is square-free. The first ten positive fundamental discriminants are: : 1, 5, 8, 12, 13, 17, 21, 24, 28, 29, 33 . The first ten negative fundamental discriminants are: : −3, −4, −7, −8, −11, −15, −19, −20, −23, −24, −31 . == Connection with quadratic fields == There is a connection between the theory of integral binary quadratic forms and the arithmetic of quadratic number fields. A basic property of this connection is that ''D''0 is a fundamental discriminant if, and only if, ''D''0 = 1 or ''D''0 is the discriminant of a quadratic number field. There is exactly one quadratic field for every fundamental discriminant ''D''0 ≠ 1, up to isomorphism. Caution: This is the reason why some authors consider 1 not to be a fundamental discriminant. One may interpret ''D''0 = 1 as the degenerated "quadratic" field Q (the rational numbers). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Fundamental discriminant」の詳細全文を読む スポンサード リンク
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