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# Talk:Unique factorization domain

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## Quadratic integer ring

Note that (what we have here)

- The elements of the quadratic integer ring

is not the same as

- The elements of the quadratic number field

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

The elements of the quadratic integer ring are of the form

while the elements of the quadratic number field are of the form

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

is a field since any nonzero element has a multiplicative inverse

while this is not the case for . — 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 with squarefree integer. — Daniel Forgues 03:19, 23 September 2012 (UTC)

- The page quadratic number fields should be moved to quadratic integer rings (or

- If I understand the notation mentioned above, (quadratic quotient fields) means the same as . — Daniel Forgues 03:19, 23 September 2012 (UTC)