mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, in particular the study of

abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...

, a Dedekind–Hasse norm is a function on an

integral domain In mathematics, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibilit ...

that generalises the notion of a

Euclidean function In mathematics, more specifically in ring theory, a Euclidean domain (also called a Euclidean ring) is an integral domain that can be endowed with a Euclidean function which allows a suitable generalization of Euclidean division of integers. Thi ...

Euclidean domain In mathematics, more specifically in ring theory, a Euclidean domain (also called a Euclidean ring) is an integral domain that can be endowed with a Euclidean function which allows a suitable generalization of Euclidean division of integers. Th ...

Definition

Let ''R'' be an integral domain and ''g'' : ''R'' → Z_≥0 be a function from ''R'' to the non-negative

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

s. Denote by 0_''R'' the additive identity of ''R''. The function ''g'' is called a ''Dedekind–Hasse norm'' on ''R'' if the following three conditions are satisfied: * ''g''(''a'') = 0

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...

''a'' = 0_''R'', * for any nonzero elements ''a'' and ''b'' in ''R'' either: ** ''b''

divides In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a '' multiple'' of m. An integer n is divisible or evenly divisibl ...

''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 in the

article. If the value of ''x'' can always be taken as 1 then ''g'' will in fact be a Euclidean function and ''R'' will therefore be a Euclidean domain.

Integral and principal ideal domains

The notion of a Dedekind–Hasse norm was developed independently by

Richard Dedekind Julius Wilhelm Richard Dedekind (; ; 6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and the axiomatic foundations of arithmetic. H ...

and, later, by

Helmut Hasse Helmut Hasse (; 25 August 1898 – 26 December 1979) 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 class field theory and ...

. They both noticed it was precisely the extra piece of structure needed to turn an integral domain into a

principal ideal domain In mathematics, a principal ideal domain, or PID, is an integral domain (that is, a non-zero commutative ring without nonzero zero divisors) in which every ideal is principal (that is, is formed by the multiples of a single element). Some author ...

. To wit, they proved that if an integral domain ''R'' has a Dedekind–Hasse norm, then ''R'' is a principal ideal domain.

Example

Let ''K'' be a field and consider the

polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, ...

''K'' 'X'' The function ''g'' on this domain that maps a nonzero

polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...

''p'' to 2^deg(''p''), where deg(''p'') is the degree of ''p'', and maps the zero polynomial to zero, is a Dedekind–Hasse norm on ''K'' 'X'' The first two conditions are satisfied simply by the definition of ''g'', while the third condition can be proved using

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 generalized version of the familiar arithmetic technique called long division. It can be done easily by hand, bec ...

References

* R. Sivaramakrishnan, ''Certain number-theoretic episodes in algebra'',

CRC Press The CRC Press, LLC is an American publishing group that specializes in producing technical books. Many of their books relate to engineering, science and mathematics. Their scope also includes books on business, forensics and information technol ...

, 2006.

External links

* {{DEFAULTSORT:Dedekind-Hasse norm Ring theory