Examples of PIDs that are not Fields
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If $R=mathbb{Z}[frac{1}{2}(1+sqrt{-19})]$ is an example of a PID which is not a Euclidean domain. But since Euclidean domains may not necessarily be fields, this doesn't say much unless $R$ in this case or in other cases is a field. Is $R$ a field? What are ways to check if a PID is a field?
What are some examples of PIDs that are not fields?
abstract-algebra ring-theory field-theory principal-ideal-domains
asked Dec 12 '18 at 20:57
Tomás PalamásTomás Palamás
363211
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add a comment |
$begingroup$
If $R=mathbb{Z}[frac{1}{2}(1+sqrt{-19})]$ is an example of a PID which is not a Euclidean domain. But since Euclidean domains may not necessarily be fields, this doesn't say much unless $R$ in this case or in other cases is a field. Is $R$ a field? What are ways to check if a PID is a field?
What are some examples of PIDs that are not fields?
abstract-algebra ring-theory field-theory principal-ideal-domains
asked Dec 12 '18 at 20:57
Tomás PalamásTomás Palamás
363211
$endgroup$
2
$begingroup$
Example: $Bbb Z$ is a PID, but not a field.
$endgroup$
– cansomeonehelpmeout
Dec 12 '18 at 21:01
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Many, many, yea, infinitely many. $k[t]$ for an indeterminate $t$ and $k$ a field. $k[[t]]$. $Bbb Z$. $Bbb Z[1/n]$ for any nonzero integer $n$.
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– Lubin
Dec 12 '18 at 21:04
2
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DaRT search for nonfield PIDs.
$endgroup$
– rschwieb
Dec 12 '18 at 21:22
$begingroup$
@rschwieb wow great resource. Thanks!
$endgroup$
– Tomás Palamás
Dec 12 '18 at 21:38
add a comment |
$begingroup$
If $R=mathbb{Z}[frac{1}{2}(1+sqrt{-19})]$ is an example of a PID which is not a Euclidean domain. But since Euclidean domains may not necessarily be fields, this doesn't say much unless $R$ in this case or in other cases is a field. Is $R$ a field? What are ways to check if a PID is a field?
What are some examples of PIDs that are not fields?
abstract-algebra ring-theory field-theory principal-ideal-domains
asked Dec 12 '18 at 20:57
Tomás PalamásTomás Palamás
363211
$endgroup$
If $R=mathbb{Z}[frac{1}{2}(1+sqrt{-19})]$ is an example of a PID which is not a Euclidean domain. But since Euclidean domains may not necessarily be fields, this doesn't say much unless $R$ in this case or in other cases is a field. Is $R$ a field? What are ways to check if a PID is a field?
What are some examples of PIDs that are not fields?
abstract-algebra ring-theory field-theory principal-ideal-domains
abstract-algebra ring-theory field-theory principal-ideal-domains
asked Dec 12 '18 at 20:57
Tomás PalamásTomás Palamás
363211
asked Dec 12 '18 at 20:57
Tomás PalamásTomás Palamás
363211
asked Dec 12 '18 at 20:57
Tomás PalamásTomás Palamás
363211
asked Dec 12 '18 at 20:57
Tomás PalamásTomás Palamás
363211
asked Dec 12 '18 at 20:57
Tomás PalamásTomás Palamás
363211
363211
2
$begingroup$
Example: $Bbb Z$ is a PID, but not a field.
$endgroup$
– cansomeonehelpmeout
Dec 12 '18 at 21:01
$begingroup$
Many, many, yea, infinitely many. $k[t]$ for an indeterminate $t$ and $k$ a field. $k[[t]]$. $Bbb Z$. $Bbb Z[1/n]$ for any nonzero integer $n$.
$endgroup$
– Lubin
Dec 12 '18 at 21:04
2
$begingroup$
DaRT search for nonfield PIDs.
$endgroup$
– rschwieb
Dec 12 '18 at 21:22
$begingroup$
@rschwieb wow great resource. Thanks!
$endgroup$
– Tomás Palamás
Dec 12 '18 at 21:38
add a comment |
2
$begingroup$
Example: $Bbb Z$ is a PID, but not a field.
$endgroup$
– cansomeonehelpmeout
Dec 12 '18 at 21:01
$begingroup$
Many, many, yea, infinitely many. $k[t]$ for an indeterminate $t$ and $k$ a field. $k[[t]]$. $Bbb Z$. $Bbb Z[1/n]$ for any nonzero integer $n$.
$endgroup$
– Lubin
Dec 12 '18 at 21:04
2
$begingroup$
DaRT search for nonfield PIDs.
$endgroup$
– rschwieb
Dec 12 '18 at 21:22
$begingroup$
@rschwieb wow great resource. Thanks!
$endgroup$
– Tomás Palamás
Dec 12 '18 at 21:38
2
2
$begingroup$
Example: $Bbb Z$ is a PID, but not a field.
$endgroup$
– cansomeonehelpmeout
Dec 12 '18 at 21:01
$begingroup$
Example: $Bbb Z$ is a PID, but not a field.
$endgroup$
– cansomeonehelpmeout
Dec 12 '18 at 21:01
$begingroup$
Many, many, yea, infinitely many. $k[t]$ for an indeterminate $t$ and $k$ a field. $k[[t]]$. $Bbb Z$. $Bbb Z[1/n]$ for any nonzero integer $n$.
$endgroup$
– Lubin
Dec 12 '18 at 21:04
$begingroup$
Many, many, yea, infinitely many. $k[t]$ for an indeterminate $t$ and $k$ a field. $k[[t]]$. $Bbb Z$. $Bbb Z[1/n]$ for any nonzero integer $n$.
$endgroup$
– Lubin
Dec 12 '18 at 21:04
2
2
$begingroup$
DaRT search for nonfield PIDs.
$endgroup$
– rschwieb
Dec 12 '18 at 21:22
$begingroup$
DaRT search for nonfield PIDs.
$endgroup$
– rschwieb
Dec 12 '18 at 21:22
$begingroup$
@rschwieb wow great resource. Thanks!
$endgroup$
– Tomás Palamás
Dec 12 '18 at 21:38
$begingroup$
@rschwieb wow great resource. Thanks!
$endgroup$
– Tomás Palamás
Dec 12 '18 at 21:38
add a comment |
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2
$begingroup$
Example: $Bbb Z$ is a PID, but not a field.
$endgroup$
– cansomeonehelpmeout
Dec 12 '18 at 21:01
$begingroup$
Many, many, yea, infinitely many. $k[t]$ for an indeterminate $t$ and $k$ a field. $k[[t]]$. $Bbb Z$. $Bbb Z[1/n]$ for any nonzero integer $n$.
$endgroup$
– Lubin
Dec 12 '18 at 21:04
2
$begingroup$
DaRT search for nonfield PIDs.
$endgroup$
– rschwieb
Dec 12 '18 at 21:22
$begingroup$
@rschwieb wow great resource. Thanks!
$endgroup$
– Tomás Palamás
Dec 12 '18 at 21:38