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

Quartic Reciprocity - Gaussian Integers - Facts and Terminology

... Gauss proves that Z is a

... 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 -

... rings, in particular rings of integers, there is a

**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

... 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 -

... of integers O of an algebraic number field K is that of the

**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

... 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)

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