Talk:Unique factorization domain

Quadratic integer ring

Note that (what we have here)

The elements of the quadratic integer ring $\scriptstyle \mathbb {Z} [{\sqrt {-5}}]\,$

is not the same as

The elements of the quadratic number field $\scriptstyle \mathbb {Q} [{\sqrt {-5}}]\,$

— Daniel Forgues 04:32, 20 September 2012 (UTC)

The elements of the quadratic integer ring $\scriptstyle \mathbb {Z} [{\sqrt {-5}}]\,$ are of the form

$m+n{\sqrt {-5}},\quad m,\,n\in \mathbb {Z} ,\,$

while the elements of the quadratic number field $\scriptstyle \mathbb {Q} [{\sqrt {-5}}]\,$ are of the form

$a+b{\sqrt {-5}},\quad a,\,b\in \mathbb {Q} .\,$

— Daniel Forgues 04:36, 20 September 2012 (UTC)

$\scriptstyle \mathbb {Q} [{\sqrt {-5}}]\,$ is a field since any nonzero element has a multiplicative inverse

${\frac {1}{a+b{\sqrt {-5}}}}={\frac {a-b{\sqrt {-5}}}{(a+b{\sqrt {-5}})(a-b{\sqrt {-5}})}}={\frac {a-b{\sqrt {-5}}}{a^{2}+5b^{2}}}\in \mathbb {Q} [{\sqrt {-5}}],\quad a,\,b\in \mathbb {Q} ,\,$

while this is not the case for $\scriptstyle \mathbb {Z} [{\sqrt {-5}}]\,$. — Daniel Forgues 04:44, 20 September 2012 (UTC)

There may be more of that confusion in the quadratic number fields page. Can you look it over, Daniel? Alonso del Arte 15:37, 20 September 2012 (UTC)

There is a notational issue worth mentioning. In the abstract A[b] denotes a ring where the base ring is extended by adjoining the element b (or the elements of b, if it is a set; yes there is an abuse of notation here but it is common and does not cause trouble). On the other hand A(b) denotes the field of fractions over A[b]. In this case Q[sqrt(-5)] "=" Q(sqrt(5)) insofar as there is a ring isomorphism between the two, but (depending on your formalism) they probably aren't literally equal as such.

Charles R Greathouse IV 23:02, 20 September 2012 (UTC)

The page quadratic number fields should be moved to quadratic integer rings (or integer subring of quadratic number fields), while quadratic number fields should be about $\scriptstyle \mathbb {Q} [{\sqrt {d}}]\,$ with $\scriptstyle d\,$ squarefree integer. — Daniel Forgues 03:19, 23 September 2012 (UTC)

If I understand the notation mentioned above, $\scriptstyle \mathbb {Z} ({\sqrt {-5}})\,$ (quadratic quotient fields) means the same as $\scriptstyle \mathbb {Q} [{\sqrt {-5}}]\,$. — Daniel Forgues 03:19, 23 September 2012 (UTC)