Seminorm on a division ring
$begingroup$
Suppose that $D$ is a division ring carrying a non-archimedian seminorm $vertcdotvert:Dtomathbb{R}$, i.e. $vert a+bvertleqmax{vert avert,vert bvert}$ and $vert abvertleq vert avertcdotvert bvert$ for all $a,bin D$, $vert 0vert=0$ and $vert 1vert=1$. Furthermore, suppose that $vert avert=0$ if and only if $a=0$.
Clearly $D$ carries a topology associated with the metric defined by $d(a,b)=vert a-bvert$, and the completion $widehat{D}$ of $D$ with respect to this topology is also a ring with a non-archimedian seminorm.
Suppose we know that $widehat{D}cong M_{n}(K)$ for some division ring $K$, $ninmathbb{N}$. Does it follow that $n=1$? If not, can anyone think of a counterexample.
abstract-algebra metric-spaces complete-spaces topological-rings division-ring
edited Dec 4 '18 at 19:26
Alex Ravsky
39.9k32282
asked Oct 22 '18 at 10:15
AdJoint-repAdJoint-rep
26718
$endgroup$
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$begingroup$
Suppose that $D$ is a division ring carrying a non-archimedian seminorm $vertcdotvert:Dtomathbb{R}$, i.e. $vert a+bvertleqmax{vert avert,vert bvert}$ and $vert abvertleq vert avertcdotvert bvert$ for all $a,bin D$, $vert 0vert=0$ and $vert 1vert=1$. Furthermore, suppose that $vert avert=0$ if and only if $a=0$.
Clearly $D$ carries a topology associated with the metric defined by $d(a,b)=vert a-bvert$, and the completion $widehat{D}$ of $D$ with respect to this topology is also a ring with a non-archimedian seminorm.
Suppose we know that $widehat{D}cong M_{n}(K)$ for some division ring $K$, $ninmathbb{N}$. Does it follow that $n=1$? If not, can anyone think of a counterexample.
abstract-algebra metric-spaces complete-spaces topological-rings division-ring
edited Dec 4 '18 at 19:26
Alex Ravsky
39.9k32282
asked Oct 22 '18 at 10:15
AdJoint-repAdJoint-rep
26718
$endgroup$
add a comment |
$begingroup$
Suppose that $D$ is a division ring carrying a non-archimedian seminorm $vertcdotvert:Dtomathbb{R}$, i.e. $vert a+bvertleqmax{vert avert,vert bvert}$ and $vert abvertleq vert avertcdotvert bvert$ for all $a,bin D$, $vert 0vert=0$ and $vert 1vert=1$. Furthermore, suppose that $vert avert=0$ if and only if $a=0$.
Clearly $D$ carries a topology associated with the metric defined by $d(a,b)=vert a-bvert$, and the completion $widehat{D}$ of $D$ with respect to this topology is also a ring with a non-archimedian seminorm.
Suppose we know that $widehat{D}cong M_{n}(K)$ for some division ring $K$, $ninmathbb{N}$. Does it follow that $n=1$? If not, can anyone think of a counterexample.
abstract-algebra metric-spaces complete-spaces topological-rings division-ring
edited Dec 4 '18 at 19:26
Alex Ravsky
39.9k32282
asked Oct 22 '18 at 10:15
AdJoint-repAdJoint-rep
26718
$endgroup$
Suppose that $D$ is a division ring carrying a non-archimedian seminorm $vertcdotvert:Dtomathbb{R}$, i.e. $vert a+bvertleqmax{vert avert,vert bvert}$ and $vert abvertleq vert avertcdotvert bvert$ for all $a,bin D$, $vert 0vert=0$ and $vert 1vert=1$. Furthermore, suppose that $vert avert=0$ if and only if $a=0$.
Clearly $D$ carries a topology associated with the metric defined by $d(a,b)=vert a-bvert$, and the completion $widehat{D}$ of $D$ with respect to this topology is also a ring with a non-archimedian seminorm.
Suppose we know that $widehat{D}cong M_{n}(K)$ for some division ring $K$, $ninmathbb{N}$. Does it follow that $n=1$? If not, can anyone think of a counterexample.
abstract-algebra metric-spaces complete-spaces topological-rings division-ring
abstract-algebra metric-spaces complete-spaces topological-rings division-ring
edited Dec 4 '18 at 19:26
Alex Ravsky
39.9k32282
asked Oct 22 '18 at 10:15
AdJoint-repAdJoint-rep
26718
edited Dec 4 '18 at 19:26
Alex Ravsky
39.9k32282
asked Oct 22 '18 at 10:15
AdJoint-repAdJoint-rep
26718
edited Dec 4 '18 at 19:26
Alex Ravsky
39.9k32282
edited Dec 4 '18 at 19:26
Alex Ravsky
39.9k32282
edited Dec 4 '18 at 19:26
Alex Ravsky
39.9k32282
39.9k32282
asked Oct 22 '18 at 10:15
AdJoint-repAdJoint-rep
26718
asked Oct 22 '18 at 10:15
AdJoint-repAdJoint-rep
26718
asked Oct 22 '18 at 10:15
AdJoint-repAdJoint-rep
26718
26718
add a comment |
add a comment |
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