GTM 84 - Ch1 - Unique Factorization

1 Results

1 Relations

\[Euclidean\ Domain\subset PID\subset UFD \]

2 PID is included in UFD

Let \(R\) be a PID and \(S\) a set of primes with the primes with the properties {A Every prime in \(R\) is associate to a prime in S. B No two primes in \(S\) are associate}. Then if \(a\in S\), \(a\not =0\), we can write:

\[a=u\prod_p p^{e(p)} \]

where \(u\) is a unit and the product is over all \(p\in S\). The unit \(u\) and the exponents \(e(p)\) are uniquely determined by \(a\). In fact, \(e(p)=ord_p\ a\)

2 Some Notes

1 K[x]: ring based on field K

• Euclidean Domain

2 In PID, irreducible element is quivalent to prime

3 \(\mathbb{Z}[i], \mathbb{Z}[\omega]\) is Euclidean Domain

4 \(\mathbb{C}[x,y,z]\) with \(x^2+y^2+z^2=1\) is a non-UFD

5 The properties of \(\mathbb{K}[x]\) is similar to \(\mathbb{Z}\)