Dedekind–Hasse norm
Encyclopedia
In mathematics
, in particular the study of abstract algebra
, a Dedekind–Hasse norm is a function on an integral domain that generalises the notion of a Euclidean function on Euclidean domain
s.
. Denote by 0R the additive identity of R. The function g is called a Dedekind–Hasse norm on R if the following three conditions are satisfied:
The third condition is a slight generalisation of condition (EF1) of Euclidean functions, as defined on the Euclidean domain
article. If the value of x can always be taken as 1 then g will in fact be a Euclidean function and R will hence be a Euclidean domain.
and, later, by Helmut Hasse
. They both noticed it was precisely the extra piece of structure needed to turn an integral domain into a principal ideal domain
. To wit, they proved that an integral domain R is a principal ideal domain if and only if
R has a Dedekind–Hasse norm.
and consider the polynomial
ring F[X]. The function g on this domain that maps a nonzero polynomial p to 2deg(p), where deg(p) is the degree of p, and maps the zero polynomial to zero, is a Dedekind–Hasse norm on F[X]. The first two conditions are satisfied simply by the definition of g, while the third condition can be proved using polynomial long division
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, in particular the study of abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, a Dedekind–Hasse norm is a function on an integral domain that generalises the notion of a Euclidean function on Euclidean domain
Euclidean domain
In mathematics, more specifically in abstract algebra and ring theory, a Euclidean domain is a ring that can be endowed with a certain structure – namely a Euclidean function, to be described in detail below – which allows a suitable generalization of the Euclidean algorithm...
s.
Definition
Let R be an integral domain and g : R → Z≥ 0 be a function from R to the non-negative rational integersInteger
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
. Denote by 0R the additive identity of R. The function g is called a Dedekind–Hasse norm on R if the following three conditions are satisfied:
- g(0R) = 0,
- if a ≠ 0R then g(a) > 0,
- for any nonzero elements a and b in R either:
- b divides a in R, or
- there exist elements x and y in R such that 0 < g(xa − yb) < g(b).
The third condition is a slight generalisation of condition (EF1) of Euclidean functions, as defined on the Euclidean domain
Euclidean domain
In mathematics, more specifically in abstract algebra and ring theory, a Euclidean domain is a ring that can be endowed with a certain structure – namely a Euclidean function, to be described in detail below – which allows a suitable generalization of the Euclidean algorithm...
article. If the value of x can always be taken as 1 then g will in fact be a Euclidean function and R will hence be a Euclidean domain.
Integral and principal ideal domains
The notion of a Dedekind–Hasse norm was developed independently by Richard DedekindRichard Dedekind
Julius Wilhelm Richard Dedekind was a German mathematician who did important work in abstract algebra , algebraic number theory and the foundations of the real numbers.-Life:...
and, later, by Helmut Hasse
Helmut Hasse
Helmut Hasse was a German mathematician working in algebraic number theory, known for fundamental contributions to class field theory, the application of p-adic numbers to local classfield theory and diophantine geometry , and to local zeta functions.-Life:He was born in Kassel, and died in...
. They both noticed it was precisely the extra piece of structure needed to turn an integral domain into a principal ideal domain
Principal ideal domain
In abstract algebra, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors refer to PIDs as...
. To wit, they proved that an integral domain R is a principal ideal domain if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
R has a Dedekind–Hasse norm.
Example
Let F be a fieldField (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
and consider the polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
ring F[X]. The function g on this domain that maps a nonzero polynomial p to 2deg(p), where deg(p) is the degree of p, and maps the zero polynomial to zero, is a Dedekind–Hasse norm on F[X]. The first two conditions are satisfied simply by the definition of g, while the third condition can be proved using polynomial long division
Polynomial long division
In algebra, polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree, a generalised version of the familiar arithmetic technique called long division...
.