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tech / sci.math / Where is Dedekind's claim that associative algebras all come out like R; C; CxR; CxC; CxCxR; CxCxC; ... ?

SubjectAuthor
* Where is Dedekind's claim that associative algebras all come out likeTimothy Golden
+- Re: Where is Dedekind's claim that associative algebras all come out-
+* Re: Where is Dedekind's claim that associative algebras all come outsergio
|`- Re: Where is Dedekind's claim that associative algebras all come outRoss A. Finlayson
`* Re: Where is Dedekind's claim that associative algebras all come outFredJeffries
 `* Re: Where is Dedekind's claim that associative algebras all come outFredJeffries
  +- Re: Where is Dedekind's claim that associative algebras all come outTimothy Golden
  `* Re: Where is Dedekind's claim that associative algebras all come outTimothy Golden
   `* Re: Where is Dedekind's claim that associative algebras all come outFredJeffries
    `- Re: Where is Dedekind's claim that associative algebras all come outTimothy Golden

1
Where is Dedekind's claim that associative algebras all come out like R; C; CxR; CxC; CxCxR; CxCxC; ... ?

<742b19c1-34c4-454f-a27b-93b85c640a65n@googlegroups.com>

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Subject: Where is Dedekind's claim that associative algebras all come out like
R; C; CxR; CxC; CxCxR; CxCxC; ... ?
From: timbandt...@gmail.com (Timothy Golden)
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 by: Timothy Golden - Fri, 7 Jan 2022 17:39 UTC

Does anyone have a link to this?

Re: Where is Dedekind's claim that associative algebras all come out like R; C; CxR; CxC; CxCxR; CxCxC; ... ?

<sr9ufd$10jd$1@gioia.aioe.org>

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Subject: Re: Where is Dedekind's claim that associative algebras all come out
like R; C; CxR; CxC; CxCxR; CxCxC; ... ?
Date: Fri, 7 Jan 2022 09:49:13 -0800
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 by: - - Fri, 7 Jan 2022 17:49 UTC

On 1/7/2022 9:39 AM, Timothy Golden wrote:
> Does anyone have a link to this?

Give it a rest you google-posting fuckwit.

Re: Where is Dedekind's claim that associative algebras all come out like R; C; CxR; CxC; CxCxR; CxCxC; ... ?

<sra23n$rt0$1@gioia.aioe.org>

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From: inva...@invalid.com (sergio)
Newsgroups: sci.math
Subject: Re: Where is Dedekind's claim that associative algebras all come out
like R; C; CxR; CxC; CxCxR; CxCxC; ... ?
Date: Fri, 7 Jan 2022 12:51:33 -0600
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 by: sergio - Fri, 7 Jan 2022 18:51 UTC

On 1/7/2022 11:39 AM, Timothy Golden wrote:
> Does anyone have a link to this?

https://plato.stanford.edu/entries/dedekind-foundations/

Re: Where is Dedekind's claim that associative algebras all come out like R; C; CxR; CxC; CxCxR; CxCxC; ... ?

<b523db6a-9465-47ba-afe2-a03cf4db81f1n@googlegroups.com>

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Subject: Re: Where is Dedekind's claim that associative algebras all come out
like R; C; CxR; CxC; CxCxR; CxCxC; ... ?
From: fredjeff...@gmail.com (FredJeffries)
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 by: FredJeffries - Fri, 7 Jan 2022 19:32 UTC

On Friday, January 7, 2022 at 9:39:19 AM UTC-8, timba...@gmail.com wrote:
> Does anyone have a link to this?

From Israel Kleiner "A History of Abstract Algebra" p 45

<quote>
In the 1860s Dedekind and Weierstrass proved that the only finite-dimensional commutative algebras over R or C, without nilpotent elements, are direct sums of copies of R or C. This means that not only addition but also multiplication in such algebras is given componentwise. This result was published only in the 1880s.
</quote>

references given are:
N. Bourbaki, Elements of the History of Mathematics, Springer-Verlag, 1994

K. H. Parshall, "H.M. Wedderburn and the structure theory of algebras", Arch. Hist. Ex. Sci.
1985, 32: 223–349.

B. L. van der Waerden, A History of Algebra, Springer-Verlag, 1985.

On pp. 268ff of her paper, Parshall gives:
Richard Dedekind, "Zur Theorie der aus n Haupteinheiten gebildeten komplexen Grossen," Göttingen Nachrichten (1885): 142

Re: Where is Dedekind's claim that associative algebras all come out like R; C; CxR; CxC; CxCxR; CxCxC; ... ?

<8f83acd4-e703-4a0a-acb4-e066f73f68c3n@googlegroups.com>

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Subject: Re: Where is Dedekind's claim that associative algebras all come out
like R; C; CxR; CxC; CxCxR; CxCxC; ... ?
From: fredjeff...@gmail.com (FredJeffries)
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 by: FredJeffries - Fri, 7 Jan 2022 21:21 UTC

On Friday, January 7, 2022 at 11:32:48 AM UTC-8, FredJeffries wrote:
> On Friday, January 7, 2022 at 9:39:19 AM UTC-8, timba...@gmail.com wrote:
> > Does anyone have a link to this?
> From Israel Kleiner "A History of Abstract Algebra" p 45
>
> <quote>
> In the 1860s Dedekind and Weierstrass proved that the only finite-dimensional commutative algebras over R or C, without nilpotent elements, are direct sums of copies of R or C. This means that not only addition but also multiplication in such algebras is given componentwise. This result was published only in the 1880s.
> </quote>
>
> references given are:
> N. Bourbaki, Elements of the History of Mathematics, Springer-Verlag, 1994
>
> K. H. Parshall, "H.M. Wedderburn and the structure theory of algebras", Arch. Hist. Ex. Sci.
> 1985, 32: 223–349.
>
> B. L. van der Waerden, A History of Algebra, Springer-Verlag, 1985.
>
> On pp. 268ff of her paper, Parshall gives:
> Richard Dedekind, "Zur Theorie der aus n Haupteinheiten gebildeten komplexen Grossen," Göttingen Nachrichten (1885): 142

https://gdz.sub.uni-goettingen.de/id/PPN235693928?tify={%22pages%22:[5],%22view%22:%22info%22}

Re: Where is Dedekind's claim that associative algebras all come out like R; C; CxR; CxC; CxCxR; CxCxC; ... ?

<b5051260-9b6c-4423-89d2-aa4b178ffa70n@googlegroups.com>

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Subject: Re: Where is Dedekind's claim that associative algebras all come out
like R; C; CxR; CxC; CxCxR; CxCxC; ... ?
From: timbandt...@gmail.com (Timothy Golden)
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 by: Timothy Golden - Sun, 9 Jan 2022 00:13 UTC

On Friday, January 7, 2022 at 4:21:53 PM UTC-5, FredJeffries wrote:
> On Friday, January 7, 2022 at 11:32:48 AM UTC-8, FredJeffries wrote:
> > On Friday, January 7, 2022 at 9:39:19 AM UTC-8, timba...@gmail.com wrote:
> > > Does anyone have a link to this?
> > From Israel Kleiner "A History of Abstract Algebra" p 45
> >
> > <quote>
> > In the 1860s Dedekind and Weierstrass proved that the only finite-dimensional commutative algebras over R or C, without nilpotent elements, are direct sums of copies of R or C. This means that not only addition but also multiplication in such algebras is given componentwise. This result was published only in the 1880s.
> > </quote>
> >
> > references given are:
> > N. Bourbaki, Elements of the History of Mathematics, Springer-Verlag, 1994
> >
> > K. H. Parshall, "H.M. Wedderburn and the structure theory of algebras", Arch. Hist. Ex. Sci.
> > 1985, 32: 223–349.
> >
> > B. L. van der Waerden, A History of Algebra, Springer-Verlag, 1985.
> >
> > On pp. 268ff of her paper, Parshall gives:
> > Richard Dedekind, "Zur Theorie der aus n Haupteinheiten gebildeten komplexen Grossen," Göttingen Nachrichten (1885): 142
> https://gdz.sub.uni-goettingen.de/id/PPN235693928?tify={%22pages%22:[5],%22view%22:%22info%22}

Nice find.

Wish I knew more German. Interesting that the quote calls for commutative algebra. I had previously thought that the crux was associative algebra.

Re: Where is Dedekind's claim that associative algebras all come out like R; C; CxR; CxC; CxCxR; CxCxC; ... ?

<58b74a80-31c4-4c1c-bfe9-b7cac0f1d81cn@googlegroups.com>

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Subject: Re: Where is Dedekind's claim that associative algebras all come out
like R; C; CxR; CxC; CxCxR; CxCxC; ... ?
From: timbandt...@gmail.com (Timothy Golden)
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 by: Timothy Golden - Wed, 9 Feb 2022 23:46 UTC

On Friday, January 7, 2022 at 4:21:53 PM UTC-5, FredJeffries wrote:
> On Friday, January 7, 2022 at 11:32:48 AM UTC-8, FredJeffries wrote:
> > On Friday, January 7, 2022 at 9:39:19 AM UTC-8, timba...@gmail.com wrote:
> > > Does anyone have a link to this?
> > From Israel Kleiner "A History of Abstract Algebra" p 45
> >
> > <quote>
> > In the 1860s Dedekind and Weierstrass proved that the only finite-dimensional commutative algebras over R or C, without nilpotent elements, are direct sums of copies of R or C. This means that not only addition but also multiplication in such algebras is given componentwise. This result was published only in the 1880s.
> > </quote>
> >
> > references given are:
> > N. Bourbaki, Elements of the History of Mathematics, Springer-Verlag, 1994
> >
> > K. H. Parshall, "H.M. Wedderburn and the structure theory of algebras", Arch. Hist. Ex. Sci.
> > 1985, 32: 223–349.
> >
> > B. L. van der Waerden, A History of Algebra, Springer-Verlag, 1985.
> >
> > On pp. 268ff of her paper, Parshall gives:
> > Richard Dedekind, "Zur Theorie der aus n Haupteinheiten gebildeten komplexen Grossen," Göttingen Nachrichten (1885): 142
> https://gdz.sub.uni-goettingen.de/id/PPN235693928?tify={%22pages%22:[5],%22view%22:%22info%22}

This is the second volume; implying there is a first volume. I wonder if the second is an overlay of the first with additions. It's 'zweiter band'.

Re: Where is Dedekind's claim that associative algebras all come out like R; C; CxR; CxC; CxCxR; CxCxC; ... ?

<8a5de510-a8a3-4dcf-b344-4faae3595ab0n@googlegroups.com>

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Subject: Re: Where is Dedekind's claim that associative algebras all come out
like R; C; CxR; CxC; CxCxR; CxCxC; ... ?
From: fredjeff...@gmail.com (FredJeffries)
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 by: FredJeffries - Thu, 10 Feb 2022 01:09 UTC

On Wednesday, February 9, 2022 at 3:46:44 PM UTC-8, timba...@gmail.com wrote:
> On Friday, January 7, 2022 at 4:21:53 PM UTC-5, FredJeffries wrote:
> > On Friday, January 7, 2022 at 11:32:48 AM UTC-8, FredJeffries wrote:
> > > On Friday, January 7, 2022 at 9:39:19 AM UTC-8, timba...@gmail.com wrote:
> > > > Does anyone have a link to this?
> > > From Israel Kleiner "A History of Abstract Algebra" p 45
> > >
> > > <quote>
> > > In the 1860s Dedekind and Weierstrass proved that the only finite-dimensional commutative algebras over R or C, without nilpotent elements, are direct sums of copies of R or C. This means that not only addition but also multiplication in such algebras is given componentwise. This result was published only in the 1880s.
> > > </quote>
> > >
> > > references given are:
> > > N. Bourbaki, Elements of the History of Mathematics, Springer-Verlag, 1994
> > >
> > > K. H. Parshall, "H.M. Wedderburn and the structure theory of algebras", Arch. Hist. Ex. Sci.
> > > 1985, 32: 223–349.
> > >
> > > B. L. van der Waerden, A History of Algebra, Springer-Verlag, 1985.
> > >
> > > On pp. 268ff of her paper, Parshall gives:
> > > Richard Dedekind, "Zur Theorie der aus n Haupteinheiten gebildeten komplexen Grossen," Göttingen Nachrichten (1885): 142
> > https://gdz.sub.uni-goettingen.de/id/PPN235693928?tify={%22pages%22:[5],%22view%22:%22info%22}
> This is the second volume; implying there is a first volume. I wonder if the second is an overlay of the first with additions. It's 'zweiter band'.

https://gdz.sub.uni-goettingen.de/volumes/id/PPN235685380

Re: Where is Dedekind's claim that associative algebras all come out like R; C; CxR; CxC; CxCxR; CxCxC; ... ?

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Subject: Re: Where is Dedekind's claim that associative algebras all come out
like R; C; CxR; CxC; CxCxR; CxCxC; ... ?
From: ross.fin...@gmail.com (Ross A. Finlayson)
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 by: Ross A. Finlayson - Thu, 10 Feb 2022 02:30 UTC

On Friday, January 7, 2022 at 10:51:45 AM UTC-8, sergio wrote:
> On 1/7/2022 11:39 AM, Timothy Golden wrote:
> > Does anyone have a link to this?
> https://plato.stanford.edu/entries/dedekind-foundations/

One time I wrote field operations for the unit radius interval
that are not the usual one.

Re: Where is Dedekind's claim that associative algebras all come out like R; C; CxR; CxC; CxCxR; CxCxC; ... ?

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Subject: Re: Where is Dedekind's claim that associative algebras all come out
like R; C; CxR; CxC; CxCxR; CxCxC; ... ?
From: timbandt...@gmail.com (Timothy Golden)
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 by: Timothy Golden - Thu, 10 Feb 2022 14:40 UTC

On Wednesday, February 9, 2022 at 8:09:28 PM UTC-5, FredJeffries wrote:
> On Wednesday, February 9, 2022 at 3:46:44 PM UTC-8, timba...@gmail.com wrote:
> > On Friday, January 7, 2022 at 4:21:53 PM UTC-5, FredJeffries wrote:
> > > On Friday, January 7, 2022 at 11:32:48 AM UTC-8, FredJeffries wrote:
> > > > On Friday, January 7, 2022 at 9:39:19 AM UTC-8, timba...@gmail.com wrote:
> > > > > Does anyone have a link to this?
> > > > From Israel Kleiner "A History of Abstract Algebra" p 45
> > > >
> > > > <quote>
> > > > In the 1860s Dedekind and Weierstrass proved that the only finite-dimensional commutative algebras over R or C, without nilpotent elements, are direct sums of copies of R or C. This means that not only addition but also multiplication in such algebras is given componentwise. This result was published only in the 1880s.
> > > > </quote>
> > > >
> > > > references given are:
> > > > N. Bourbaki, Elements of the History of Mathematics, Springer-Verlag, 1994
> > > >
> > > > K. H. Parshall, "H.M. Wedderburn and the structure theory of algebras", Arch. Hist. Ex. Sci.
> > > > 1985, 32: 223–349.
> > > >
> > > > B. L. van der Waerden, A History of Algebra, Springer-Verlag, 1985.
> > > >
> > > > On pp. 268ff of her paper, Parshall gives:
> > > > Richard Dedekind, "Zur Theorie der aus n Haupteinheiten gebildeten komplexen Grossen," Göttingen Nachrichten (1885): 142
> > > https://gdz.sub.uni-goettingen.de/id/PPN235693928?tify={%22pages%22:[5],%22view%22:%22info%22}
> > This is the second volume; implying there is a first volume. I wonder if the second is an overlay of the first with additions. It's 'zweiter band'..
> https://gdz.sub.uni-goettingen.de/volumes/id/PPN235685380

Geeze I really missed that one.
Got the erster band!

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