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Talk:Unique factorization domain

Note that (what we have here)

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

is not the same as

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

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

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

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

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

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

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

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

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

while this is not the case for ${\displaystyle \scriptstyle \mathbb {Z} [{\sqrt {-5}}]\,}$. — 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 ${\displaystyle \scriptstyle \mathbb {Q} [{\sqrt {d}}]\,}$ with ${\displaystyle \scriptstyle d\,}$ squarefree integer. — Daniel Forgues 03:19, 23 September 2012 (UTC)
If I understand the notation mentioned above, ${\displaystyle \scriptstyle \mathbb {Z} ({\sqrt {-5}})\,}$ (quadratic quotient fields) means the same as ${\displaystyle \scriptstyle \mathbb {Q} [{\sqrt {-5}}]\,}$. — Daniel Forgues 03:19, 23 September 2012 (UTC)