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Arturo Magidin science forum Guru
Joined: 25 Mar 2005
Posts: 1838

Posted: Thu Jun 29, 2006 10:28 pm Post subject:
Re: Something not computing in Langs Algebra



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 

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Lee Rudolph science forum Guru
Joined: 28 Apr 2005
Posts: 566

Posted: Thu Jun 29, 2006 10:19 pm Post subject:
Re: Something not computing in Langs Algebra



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 

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Arturo Magidin science forum Guru
Joined: 25 Mar 2005
Posts: 1838

Posted: Thu Jun 29, 2006 9:30 pm Post subject:
Re: Something not computing in Langs Algebra



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 

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Gene Ward Smith science forum Guru
Joined: 08 Jul 2005
Posts: 409

Posted: Thu Jun 29, 2006 7:33 pm Post subject:
Re: Something not computing in Langs Algebra



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. 

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Axel Vogt science forum addict
Joined: 03 May 2005
Posts: 93

Posted: Thu Jun 29, 2006 7:03 pm Post subject:
Re: Something not computing in Langs Algebra



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 

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Martin1157 science forum beginner
Joined: 10 Jun 2005
Posts: 49

Posted: Thu Jun 29, 2006 1:21 pm Post subject:
Re: Something not computing in Langs Algebra



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 

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Pubkeybreaker science forum Guru
Joined: 24 Mar 2005
Posts: 333

Posted: Thu Jun 29, 2006 1:18 pm Post subject:
Re: Something not computing in Langs Algebra



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........ 

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Martin1157 science forum beginner
Joined: 10 Jun 2005
Posts: 49

Posted: Thu Jun 29, 2006 1:13 pm Post subject:
Something not computing in Langs Algebra



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 

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