FAQFAQ   SearchSearch   MemberlistMemberlist   UsergroupsUsergroups 
 ProfileProfile   PreferencesPreferences   Log in to check your private messagesLog in to check your private messages   Log inLog in 
Forum index » Science and Technology » Math
Something not computing in Langs Algebra
Post new topic   Reply to topic Page 1 of 1 [8 Posts] View previous topic :: View next topic
Author Message
Arturo Magidin
science forum Guru


Joined: 25 Mar 2005
Posts: 1838

PostPosted: Thu Jun 29, 2006 10:28 pm    Post subject: Re: Something not computing in Langs Algebra Reply with quote

In article <e81jl6$t7v$1@panix2.panix.com>,
Lee Rudolph <lrudolph@panix.com> wrote:
Quote:
magidin@math.berkeley.edu (Arturo Magidin) writes:

In article <1151586816.023596.204670@p79g2000cwp.googlegroups.com>,
Martin <martmol@spray.se> wrote:
According to Langs Algebra (p 90 third edition):

K is a field that has characteristics 0. Then "K contains as a subfield
an isomorphic image of the rational numbers".

I have no idea how this could be correct, because neither Z or nor its
subfields (does Z have any subfields?)

No, among other reasons because Z is not a field.

This sentence seems to imply that you believe the general proposition
that "If X is not a field, then for no Y is Y a subfield of X."

That was not my intention (though it may perhaps be read that way). On
the other hand, "if X is a field, there exists a Y such that Y is a
subfield of X" is of course true. So,you will agree that a necessary
condition for something to have no subfields is that it itself is not
a field; I gave one of these necessary conditions, though I can see
why my phrasing might seem like I am giving one of (several possible)
sufficient ones.

The reason for my brevity was that it was clear that the OP was going
astray by somehow not realizing that Z is not a field, and that
reminding him of this would suffice to address his query.



--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================

Arturo Magidin
magidin@math.berkeley.edu
Back to top
Lee Rudolph
science forum Guru


Joined: 28 Apr 2005
Posts: 566

PostPosted: Thu Jun 29, 2006 10:19 pm    Post subject: Re: Something not computing in Langs Algebra Reply with quote

magidin@math.berkeley.edu (Arturo Magidin) writes:

Quote:
In article <1151586816.023596.204670@p79g2000cwp.googlegroups.com>,
Martin <martmol@spray.se> wrote:
According to Langs Algebra (p 90 third edition):

K is a field that has characteristics 0. Then "K contains as a subfield
an isomorphic image of the rational numbers".

I have no idea how this could be correct, because neither Z or nor its
subfields (does Z have any subfields?)

No, among other reasons because Z is not a field.

This sentence seems to imply that you believe the general proposition
that "If X is not a field, then for no Y is Y a subfield of X." Using
words so as to make that proposition true is certainly fair enough, but
I'm not sure it's productive. Assuming that you aren't one of the
people who call the quaternions H a field (as contrasted with a "skew
field" or "sfield" or what have you), I ask you, do you *really* want
to make it incorrect to say "H has a unique subfield isomorphic to R,
and infinitely many subfields isomorphic to C"? Would you *really*
prefer to say something like "The ring H has a unique subring which
is the image, by the forgetful functor from Rings to Fields, of a
field isomorphic to R", and so on? (If you *do* call H a field, then
it's obvious how to make more contorted examples. Hell, M(n,R) will
do, eh?)

Lee Rudolph
Back to top
Arturo Magidin
science forum Guru


Joined: 25 Mar 2005
Posts: 1838

PostPosted: Thu Jun 29, 2006 9:30 pm    Post subject: Re: Something not computing in Langs Algebra Reply with quote

In article <1151586816.023596.204670@p79g2000cwp.googlegroups.com>,
Martin <martmol@spray.se> wrote:
Quote:
According to Langs Algebra (p 90 third edition):

K is a field that has characteristics 0. Then "K contains as a subfield
an isomorphic image of the rational numbers".

I have no idea how this could be correct, because neither Z or nor its
subfields (does Z have any subfields?)

No, among other reasons because Z is not a field.


--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================

Arturo Magidin
magidin@math.berkeley.edu
Back to top
Gene Ward Smith
science forum Guru


Joined: 08 Jul 2005
Posts: 409

PostPosted: Thu Jun 29, 2006 7:33 pm    Post subject: Re: Something not computing in Langs Algebra Reply with quote

Martin wrote:
Quote:
According to Langs Algebra (p 90 third edition):

K is a field that has characteristics 0. Then "K contains as a subfield
an isomorphic image of the rational numbers".

I have no idea how this could be correct, because neither Z or nor its
subfields (does Z have any subfields?) can be isomorphic with Q, but it
certainly have characteristics 0. Could it be that he means that K
contains as a subfield an isomorphic image of a_subfield_contained_by
the rational numbers?

Z isn't a field, so it isn't a subfield.
Back to top
Axel Vogt
science forum addict


Joined: 03 May 2005
Posts: 93

PostPosted: Thu Jun 29, 2006 7:03 pm    Post subject: Re: Something not computing in Langs Algebra Reply with quote

Martin wrote:
Quote:

According to Langs Algebra (p 90 third edition):

K is a field that has characteristics 0. Then "K contains as a subfield
an isomorphic image of the rational numbers".

I have no idea how this could be correct, because neither Z or nor its
subfields (does Z have any subfields?) can be isomorphic with Q, but it
certainly have characteristics 0. Could it be that he means that K
contains as a subfield an isomorphic image of a_subfield_contained_by
the rational numbers?

Best Regards
//Martin

you always have a ring morphism Z -> R, so K contains a copy of Z
and inverting elements gives you what is meant
Back to top
Martin1157
science forum beginner


Joined: 10 Jun 2005
Posts: 49

PostPosted: Thu Jun 29, 2006 1:21 pm    Post subject: Re: Something not computing in Langs Algebra Reply with quote

Pubkeybreaker wrote:
Quote:
Martin wrote:
According to Langs Algebra (p 90 third edition):

K is a field that has characteristics 0. Then "K contains as a subfield
an isomorphic image of the rational numbers".

I have no idea how this could be correct, because neither Z or nor its
subfields (does Z have any subfields?)

Z is not a field........


Thx

of course it isn't..

//Martin
Back to top
Pubkeybreaker
science forum Guru


Joined: 24 Mar 2005
Posts: 333

PostPosted: Thu Jun 29, 2006 1:18 pm    Post subject: Re: Something not computing in Langs Algebra Reply with quote

Martin wrote:
Quote:
According to Langs Algebra (p 90 third edition):

K is a field that has characteristics 0. Then "K contains as a subfield
an isomorphic image of the rational numbers".

I have no idea how this could be correct, because neither Z or nor its
subfields (does Z have any subfields?)

Z is not a field........
Back to top
Martin1157
science forum beginner


Joined: 10 Jun 2005
Posts: 49

PostPosted: Thu Jun 29, 2006 1:13 pm    Post subject: Something not computing in Langs Algebra Reply with quote

According to Langs Algebra (p 90 third edition):

K is a field that has characteristics 0. Then "K contains as a subfield
an isomorphic image of the rational numbers".

I have no idea how this could be correct, because neither Z or nor its
subfields (does Z have any subfields?) can be isomorphic with Q, but it
certainly have characteristics 0. Could it be that he means that K
contains as a subfield an isomorphic image of a_subfield_contained_by
the rational numbers?


Best Regards
//Martin
Back to top
Google

Back to top
Display posts from previous:   
Post new topic   Reply to topic Page 1 of 1 [8 Posts] View previous topic :: View next topic
The time now is Tue Oct 23, 2018 12:33 am | All times are GMT
Forum index » Science and Technology » Math
Jump to:  

Similar Topics
Topic Author Forum Replies Last Post
No new posts Help with Boolean Algebra Questions!!! mcbivens@umes.edu Math 0 Thu Sep 14, 2006 6:58 pm
No new posts Help with Boolean Algebra Questions!!! mcbivens@umes.edu Math 0 Thu Sep 14, 2006 6:58 pm
No new posts noncommutative algebra book recommendation? adomplayer@yahoo.com Math 2 Wed Jul 19, 2006 9:35 pm
No new posts Correction factor in computing exp()? lcw1964 num-analysis 2 Sun Jul 16, 2006 7:22 am
No new posts Linear algebra txtbk recommendations... Snis Pilbor Math 1 Sat Jul 15, 2006 11:40 pm

Copyright © 2004-2005 DeniX Solutions SRL
Other DeniX Solutions sites: Electronics forum |  Medicine forum |  Unix/Linux blog |  Unix/Linux documentation |  Unix/Linux forums  |  send newsletters
 


Powered by phpBB © 2001, 2005 phpBB Group
[ Time: 0.0160s ][ Queries: 20 (0.0027s) ][ GZIP on - Debug on ]