In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a fundamental discriminant ''D'' is an
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
invariant
Invariant and invariance may refer to:
Computer science
* Invariant (computer science), an expression whose value doesn't change during program execution
** Loop invariant, a property of a program loop that is true before (and after) each iteratio ...
in the theory of
integral
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
binary quadratic form
In mathematics, a binary quadratic form is a quadratic homogeneous polynomial in two variables
: q(x,y)=ax^2+bxy+cy^2, \,
where ''a'', ''b'', ''c'' are the coefficients. When the coefficients can be arbitrary complex numbers, most results are ...
s. If is a
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to a ...
with integer coefficients, then is the
discriminant
In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the origi ...
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.
There are explicit
congruence conditions that give the set of fundamental discriminants. Specifically, ''D'' is a fundamental discriminant
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicondi ...
one of the following statements holds
* ''D'' ≡ 1 (mod 4) and is
square-free {{no footnotes, date=December 2015
In mathematics, a square-free element is an element ''r'' of a unique factorization domain ''R'' that is not divisible by a non-trivial square. This means that every ''s'' such that s^2\mid r is a unit of ''R''.
A ...
,
* ''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
In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the origi ...
of a quadratic number field. There is exactly one quadratic field for every fundamental discriminant ''D''
0 ≠ 1, up to
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
. This is the reason why some authors consider 1 not to be a fundamental discriminant, although one may interpret ''D''
0 = 1 as the discriminant of the quadratic algebra consisting of two copies of the rational field.
Factorization
Fundamental discriminants may also be characterized by their
factorization into positive and negative prime powers. Define the set
:
where the
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s congruent to 1 mod 4 are positive and those congruent to 3 mod 4 are negative. Then, a number ''D''
0 ≠ 1 is a fundamental discriminant if, and only if, it is the product of
pairwise relatively prime
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
members of ''S''.
References
*
*
* {{cite book , author=Don Zagier , authorlink=Don Zagier , title=Zetafunktionen und quadratische Körper , publisher=
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
Originally founded in 1842 in ...
, location=Berlin, New York , isbn=978-3-540-10603-6 , year = 1981
See also
*
Quadratic integer
In number theory, quadratic integers are a generalization of the usual integers to quadratic fields. Quadratic integers are algebraic integers of degree two, that is, solutions of equations of the form
:
with and (usual) integers. When algebrai ...
Algebraic number theory