# Unique Factorization

### Some articles on unique factorization, factorization, unique:

Quartic Reciprocity - Gaussian Integers - Facts and Terminology
... Gauss proves that Z is a unique factorization domain and shows that the primes fall into three classes 2 is a special case 2 = i3 (1 + i)2 ... and a factorization of the split primes is 5 = (2 + i) × (2 − i), 13 = (2 + 3i) × (2 − 3i), 17 = (4 + i) × (4 − i), 29 = (2 + 5i) × (2 − 5i).. ... In order to state the unique factorization theorem, it is necessary to have a way of distinguishing one of the associates of a number ...
Algebraic Number Field - Algebraicity and Ring of Integers - Unique Factorization and Class Number
... rings, in particular rings of integers, there is a unique factorization of ideals into a product of prime ideals ... a proper extension of Q need not admit unique factorization of numbers into a product of prime numbers or, more precisely, prime elements ... integers, for example in OQ(√−5) = Z, the uniqueness of the factorization fails 6 = 2 · 3 = (1 + √−5) · (1 − √−5) ...
Irreducible Polynomial
... that equally apply to irreducible polynomials, such as the essentially unique factorization into prime or irreducible factors Every polynomial with coefficients in a field F can be factorized into polynomials that ... This factorization is unique up to permutation of the factors and the multiplication of the factors by nonzero constants from F ... This property of unique factorization is commonly expressed by saying that the polynomial rings over a field are unique factorization domains ...
Algebraic Number Theory - Basic Notions - Unique Factorization and The Ideal Class Group
... of integers O of an algebraic number field K is that of the unique factorization of integers into prime numbers ... The prime numbers in Z are generalized to irreducible elements in O, and though the unique factorization of elements of O into irreducible elements may hold in some cases (such ...
Cyclotomic Field - Relation With Fermat's Last Theorem
... If unique factorization of algebraic integers were true, then it could have been used to rule out the existence of nontrivial solutions to Fermat's equation ... Unfortunately, the unique factorization fails in general – for example, for n = 23 – but Kummer found a way around this difficulty ... for the prime numbers in the cyclotomic field Q(ζp), expressed the failure of unique factorization quantitatively via the class number hp and ...

### Famous quotes containing the word unique:

The unique and supreme voluptuousness of love lies in the certainty of committing evil. And men and women know from birth that in evil is found all sensual delight.
Charles Baudelaire (1821–1867)