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Lang's Algebra

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Nobuo Saito

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Apr 27, 2002, 9:07:57 AM4/27/02
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I saw Lang's book "Algebra"(new edition) in a bookstore in Japan.
It was about $120!
Before I buy it, I'd like to hear opinions of others about his book.
Thanks.

Steve Liem

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Apr 27, 2002, 9:21:07 AM4/27/02
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A very good Algebra resource.

"Nobuo Saito" <genki...@hotmail.com> wrote in message
news:c1d437f.02042...@posting.google.com...

Arturo Magidin

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Apr 27, 2002, 12:37:22 PM4/27/02
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In article <c1d437f.02042...@posting.google.com>,

Lang is a very idiosyncratic author. Some people love his books, others
hate them.

His _Algebra_ is encyclopedic. It contains practically everything you
might ever see in a first year graduate course in Algebra, plus much
more on top of it. Most people, however, find it a hard book to learn
from.

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

Arturo Magidin
mag...@math.berkeley.edu

Jesse F. Hughes

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Apr 27, 2002, 2:15:46 PM4/27/02
to
mag...@math.berkeley.edu (Arturo Magidin) writes:

> In article <c1d437f.02042...@posting.google.com>,
> Nobuo Saito <genki...@hotmail.com> wrote:
> >I saw Lang's book "Algebra"(new edition) in a bookstore in Japan.
> >It was about $120!
> >Before I buy it, I'd like to hear opinions of others about his book.
>
> Lang is a very idiosyncratic author. Some people love his books, others
> hate them.
>
> His _Algebra_ is encyclopedic. It contains practically everything you
> might ever see in a first year graduate course in Algebra, plus much
> more on top of it. Most people, however, find it a hard book to learn
> from.

Both Arturo and Steve Liem make points with which I agree. The book
is an excellent reference, but can be difficult as a first look at
algebra. I suspect that there are more occurrences of words like,
"clearly," "obviously," and "trivially" in that text than in any other
book.

--
Jesse Hughes

"Metaphysicians are musicians without musical ability."
--Rudolf Carnap, "The Elimination of Metaphysics..."

Marc Olschok

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Apr 29, 2002, 8:39:35 AM4/29/02
to

I think it is a very nice book. Although I can only tell from the
first edition (the small Addison-Wesley).

I wonder where the $120 come from. As far as I know the
Springer 3rd ed (which seems to be a reissue of a former 3rd ed)
costs about EUR 74,95 (ISBN 0-387-95385-X), which should be approx. $65.

If you are in doubt about the book, you should probably check it out
at your library. As I mentioned above, there has been a 3rd edition
already for a couple of years, and I do not think that there are a
lot of differences.

best regards, Marc.

Chan-Ho Suh

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Apr 29, 2002, 3:33:59 PM4/29/02
to

"Nobuo Saito" <genki...@hotmail.com> wrote in message
news:c1d437f.02042...@posting.google.com...

I would go back to the bookstore and look in a section containing material
you already know well. See if you can make heads or tails of what Lang
wrote, and make your decision accordingly. It is a pretty good reference
according to quite a few algebraists. I think you can get it a lot cheaper
though.

I recall that Ken Ribet used to have on his webpage a story about how when
Lang once came to visit Berkeley he was put in Ribet's office. Apparently
Lang had occasion to refer to his own text, which of course Ribet had. Now
Ribet had written in the book, right after the part in the introduction that
Lang writes that minimal background is needed to understand the book,
something like "pretentious old fart" or a similar thing. When Ribet comes
back from vacation and opens up his Lang's Algebra he finds "fuck you!" had
been written under his comment.

Unfortunately, Ribet has taken down that link; maybe that has something to
do with Lang's recent/current visit to Berkeley.


Nobuo Saito

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Apr 29, 2002, 5:33:13 PM4/29/02
to
Marc Olschok <sa7...@l1-hrz.uni-duisburg.de> wrote in message news:<aajeu7$l0i$1...@a1-hrz.uni-duisburg.de>...

Of course, $120 includes shipment.
But Japanese bookstores of foreign books are notorious in their prices.

Thank you and the other people who kindly gave me information about the book.
N.S.

Dik T. Winter

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Apr 29, 2002, 7:45:45 PM4/29/02
to
In article <aak6r6$aa7$1...@persian.noc.ucla.edu> "Chan-Ho Suh" <cs...@math.ucla.edu> writes:
> "Nobuo Saito" <genki...@hotmail.com> wrote in message
> news:c1d437f.02042...@posting.google.com...
> > I saw Lang's book "Algebra"(new edition) in a bookstore in Japan.
> > It was about $120!
>
> I think you can get it a lot cheaper
> though.

Perhaps. It is $70 at Amazon. However I have no idea what shipping and
duty for Japan might be. Perhaps enough to make $120 a bargain. I once
ordered a game from the US. Price about $50. Shipping, handling and
duty mounted up to about $70. But I just checked Amazon Japan. They
sell it for 9,232 Yen, which is also about $70 (although I have no idea
how fast *they* ship, I could not read that).
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/

John Leo

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Apr 30, 2002, 12:31:54 AM4/30/02
to
Chan-Ho Suh writes:

>I recall that Ken Ribet used to have on his webpage a story about how when
>Lang once came to visit Berkeley he was put in Ribet's office. Apparently
>Lang had occasion to refer to his own text, which of course Ribet had. Now
>Ribet had written in the book, right after the part in the introduction that
>Lang writes that minimal background is needed to understand the book,
>something like "pretentious old fart" or a similar thing. When Ribet comes
>back from vacation and opens up his Lang's Algebra he finds "fuck you!" had
>been written under his comment.

I believe the exact phrase was "pretentious asshole". I thought I had saved the scanned images but can't seem
to locate them. I took Ribet's course last Fall through Berkeley extension and it was quite an experience. I
found myself agreeing with Ribet's comment--Lang's attitude seems almost adversarial, as if he's saying "if you
were as smart as me, you wouldn't need to read my book in the first place!" It's the worst math text I've ever
tried to read--very sloppily written. George Bergman at Berleley has written a supplement to the book which I
believe is essential if you really want to try to read it as a text--at the minimum you can realize you're not
stupid when Lang says something that seems to make no sense at all, and in fact doesn't.

I made many efforts to read Lang, but I found it just too much effort with almost no reward. I'd fall back on
Dummit and Foote, a beautifully written book and my favorite algebra text, and find they presented the same
material incomparably better. My impression was that most or all other students in the class also disliked or
hated Lang and supplemented it with either Hungerford or D&F. Although D&F is not normally considered a grad
algebra text I found it covered most of the material we did in class, and I hope more grad algebra classes will
use it, perhaps with supplements for category theory and more advanced topics.

Once I understood the material I would look back at Lang and found I could follow it better the second time, so
it may be of value for someone looking for a different perspective for already learned material, but even then I
have doubts. Probably someone with the right attitude and a high level of sophistication will appreciate it
more--from Nobuo's posts I would guess he's such a person.

Ribet apparently likes the book now, and said one reason he uses it is because the exercises are so wonderful.
I'd agree there are a lot of excellent exercises in it, but many of them are very hard, and often didn't seem to
be the most efficient way to learn the material. I wished we'd had a mix of easier ones to go with them.

>Unfortunately, Ribet has taken down that link; maybe that has something to
>do with Lang's recent/current visit to Berkeley.

The incident with the book happened long ago; Ribet and Lang seem to be good friends now, and Lang regularly
visits Berkeley. In fact he visited our class the first day, and Ribet greeted him by asking what his favorite
homomorphism was. To see the answer look at the end of the solutions to problem set 2 posted on Ribet's 250A
homepage.

Incidentally Chan-Ho, I will be starting in the PhD program at UCLA this Fall. I look forward to meeting you.

John leo
http://www.halfaya.org/leo/

Nobuo Saito

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Apr 30, 2002, 7:51:10 AM4/30/02
to
"Chan-Ho Suh" <cs...@math.ucla.edu> wrote in message news:<aak6r6$aa7$1...@persian.noc.ucla.edu>...

> "Nobuo Saito" <genki...@hotmail.com> wrote in message
> news:c1d437f.02042...@posting.google.com...
> > I saw Lang's book "Algebra"(new edition) in a bookstore in Japan.
> > It was about $120!
> > Before I buy it, I'd like to hear opinions of others about his book.
> > Thanks.
>
> I would go back to the bookstore and look in a section containing material
> you already know well. See if you can make heads or tails of what Lang
> wrote, and make your decision accordingly. It is a pretty good reference
> according to quite a few algebraists. I think you can get it a lot cheaper
> though.
>

Actually I found it interesting. When I read a section on places
and valuation rings in his book, I thought the exposition was similar
to my articles(Prerequisites for Hartshorne's Algebraic Geometry) here
in sci.math, though I think mine is easier and more detailed than his
(advertisement mode). :-)
Also the book contained elimination theory which I am planning to include
in my articles, too.
And there are topics in the book I don't know.

Nobuo Saito

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Apr 30, 2002, 7:55:14 AM4/30/02
to
"Dik T. Winter" <Dik.W...@cwi.nl> wrote in message news:<GvCso...@cwi.nl>...

> In article <aak6r6$aa7$1...@persian.noc.ucla.edu> "Chan-Ho Suh" <cs...@math.ucla.edu> writes:
> > "Nobuo Saito" <genki...@hotmail.com> wrote in message
> > news:c1d437f.02042...@posting.google.com...
> > > I saw Lang's book "Algebra"(new edition) in a bookstore in Japan.
> > > It was about $120!
> >
> > I think you can get it a lot cheaper
> > though.
>
> Perhaps. It is $70 at Amazon. However I have no idea what shipping and
> duty for Japan might be. Perhaps enough to make $120 a bargain. I once
> ordered a game from the US. Price about $50. Shipping, handling and
> duty mounted up to about $70. But I just checked Amazon Japan. They
> sell it for 9,232 Yen, which is also about $70 (although I have no idea
> how fast *they* ship, I could not read that).

Thank you for the infomation.
Maybe I'll try Amazon Japan.
It was about 13,500 Yen in the bookstore.

Achava Nakhash, the Loving Snake

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Apr 30, 2002, 12:58:31 PM4/30/02
to
John Leo <l...@halfaya.org> wrote in message news:<3CCE1EF8...@halfaya.org>...
> Chan-Ho Suh writes:

> > I believe the exact phrase was "pretentious asshole". I thought I had saved the scanned images but can't seem
> to locate them. I took Ribet's course last Fall through Berkeley extension and it was quite an experience. I
> found myself agreeing with Ribet's comment--Lang's attitude seems almost adversarial, as if he's saying "if you
> were as smart as me, you wouldn't need to read my book in the first place!" It's the worst math text I've ever
> tried to read--very sloppily written. George Bergman at Berleley has written a supplement to the book which I
> believe is essential if you really want to try to read it as a text--at the minimum you can realize you're not
> stupid when Lang says something that seems to make no sense at all, and in fact doesn't.
>
> I made many efforts to read Lang, but I found it just too much effort with almost no reward. I'd fall back on
> Dummit and Foote, a beautifully written book and my favorite algebra text, and find they presented the same
> material incomparably better. My impression was that most or all other students in the class also disliked or
> hated Lang and supplemented it with either Hungerford or D&F. Although D&F is not normally considered a grad
> algebra text I found it covered most of the material we did in class, and I hope more grad algebra classes will
> use it, perhaps with supplements for category theory and more advanced topics.
>
> Once I understood the material I would look back at Lang and found I could follow it better the second time, so
> it may be of value for someone looking for a different perspective for already learned material, but even then I
> have doubts. Probably someone with the right attitude and a high level of sophistication will appreciate it
> more--from Nobuo's posts I would guess he's such a person.
>
> Ribet apparently likes the book now, and said one reason he uses it is because the exercises are so wonderful.
> I'd agree there are a lot of excellent exercises in it, but many of them are very hard, and often didn't seem to
> be the most efficient way to learn the material. I wished we'd had a mix of easier ones to go with them.
>

> The incident with the book happened long ago; Ribet and Lang seem to be good friends now, and Lang regularly
> visits Berkeley. In fact he visited our class the first day, and Ribet greeted him by asking what his favorite
> homomorphism was. To see the answer look at the end of the solutions to problem set 2 posted on Ribet's 250A
> homepage.
>

My experience with the Lang Algebra book is that it is a treasure
trove of excellent material and exercises, but that it is very
difficult to read. As others have said, it becomes easier and more
useful when you have studied the material from more accessible
sources, and then it takes you past them. The story I heard around
1970 was that Lang would rent a hotel room for a weekend and produce
the entire book. It does contain many subtle errors that could easily
have been the result of an over-hurried creation. I am obviously
talking about the first edition. The great advantage of Lang's
Algebra as well as his other books is that he has a definite point of
view about how to look at the subject under discussion. It is usually
the "right" point of view for a lot of applications and extensions.
Lang is a famously crusty character, but I have some evidence
that he is a nice guy with a weird sense of humor and an almost
unbelievable level of intensity. I met him in the mid seventies when
I was a graduate student at Berkelely and he was on one of his many
visits. I was attempting to study class-field theory with a German
guy who was there for the year, and we got stuck on a point and
decided to ask the real expert. Lang was practically vibrating, but
he answered our question nicely enough. During a colloquium
presentation that he gave, he through chalk at a guy who asked the
same stupid question twice and told him to go study calculus or some
such, confirming several stories we had already heard about his chalk
throwing habit. On another occasion Singer walked in late to a talk
Lang was giving and said that something like, "The right way to look
at what you are doing is ..." Lang's reply, rather than thank you for
that insight was something like, "What kind of goddamned Nazi are you
coming in and telling me how I should look at things ..." I can't
imagine anyone but Lang coming up with that sort of snappy reply.
After the talk, the two of them were discussing what Singer said as if
they were old friends of long standing, and they probably were, so it
was just playfulness of a weird sort. In 1972 I went back to Yale to
visit some friends and saw Lang playing frisbee with some of the
undergraduates. He looked a bit incongrous, a guy who appeared to be
in his fifties, and quite famous at that, running around playing like
a kid with the college students. Now that I am a similar age, I also
appreciate the physical demand it must have been.
Well, enough of these reminiscenses.

Regards,
Achava

ayatollah potassium

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Apr 30, 2002, 5:31:02 PM4/30/02
to

Nobuo Saito wrote:

all props to lang the person and lang the author, but his book tries
too hard to be encyclopedic, and is old fashioned. too much purity
of method, not enough overview and relation with geometry
and topology.

Michael Barr

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Apr 30, 2002, 7:57:30 PM4/30/02
to
"Dik T. Winter" <Dik.W...@cwi.nl> wrote in message news:<GvCso...@cwi.nl>...
> In article <aak6r6$aa7$1...@persian.noc.ucla.edu> "Chan-Ho Suh" <cs...@math.ucla.edu> writes:
> > "Nobuo Saito" <genki...@hotmail.com> wrote in message
> > news:c1d437f.02042...@posting.google.com...
> > > I saw Lang's book "Algebra"(new edition) in a bookstore in Japan.
> > > It was about $120!
> >
> > I think you can get it a lot cheaper
> > though.
>
> Perhaps. It is $70 at Amazon. However I have no idea what shipping and
> duty for Japan might be. Perhaps enough to make $120 a bargain. I once
> ordered a game from the US. Price about $50. Shipping, handling and
> duty mounted up to about $70. But I just checked Amazon Japan. They
> sell it for 9,232 Yen, which is also about $70 (although I have no idea
> how fast *they* ship, I could not read that).


Exercise: Take any proof from Lang and simplify it and make it
clearer at the same time.

Although probably not literally doable, it is for an awful lot of
them. Yet I always came back to it when I taught algebra because the
selection of topics was so good. The most interesting, and maybe the
most difficult, chapter is on the real Nullstellensatz and Artin's
solution to the Hilbert problem of writing a positive rational
function as a sum of squares.

John Leo

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Apr 30, 2002, 8:00:30 PM4/30/02
to
"Achava Nakhash, the Loving Snake" wrote:

> [...] Lang is a famously crusty character, but I have some evidence


> that he is a nice guy with a weird sense of humor and an almost

> unbelievable level of intensity. [...]

Thanks very much for the Lang stories; it's always fun to hear anecdotes
about mathematicians, and Lang is a particularly interesting character.
When he visited Ribet's class (just for a couple minutes) he seemed
quite meek and mild-mannered. I had trouble believing he was the same
person who had written our textbook, not to mention has recently been
fighting against tenure given to other Yale faculty

(http://www.yaleherald.com/archive/xxix/2000.02.11/opinion/p10lang.html
http://www.yaleherald.com/archive/xxix/2000.02.11/news/p6lang.html)
and against the the theory that HIV causes AIDS (see
http://www.virusmyth.net/aids/data/slquestions.htm
and his bizzare "acceptance" of the 1999 Steele Prize (for mathematical
exposition no less!):
http://www.ams.org/notices/199904/comm-steele-prz.pdf)

It's especially interesting that he used to throw chalk at students. In
"Fine Hall in its Golden Age" Rota (one of my teachers at MIT) talks
about how Emil Artin used to throw chalk at his students, and also how
his grad students (of whom Lang was one) liked to emulate him. So I
guess that's where he learned it! Nasar's "A Beautiful Mind" also has
an interesting Lang anecdote (p. 73):
<<
On one occasion, Nash was baiting one of Artin's students by telling him
that the best way into Artin's graces was to catch his beautiful daugher
Karin. The student, Serge Lang, who everyone knew was painfully
obsessed by his shyness around girls, threw a cup of hot tea in Nash's
face. Nash chased him around the table, threw him to the ground, and
stuffed ice cubes down the back of his shirt.
>>

I believe Karin Artin eventually married John Tate, another of Artin's
students! I don't know if Tate threw chalk as well but Tate was the
advisor of Ken Ribet, who certainly does not. Ribet was in fact
particularly kind about answering student questions, including mine.
Emil Artin's son Mike Artin (who taught me algebra at MIT) was also a
very nice guy and excellent lecturer.

I'm starting to have some doubts about my memory of Ribet's note in
Lang's book. Maybe it was "pompous ass". That sounds more likely. I
guess he removed it from the web page just because it's not something
you'd leave up permanently, as funny as the story is. The problem is
that the story now risks getting distorted in the retelling....

John Leo
http://www.halfaya.org/leo/

Dik T. Winter

unread,
Apr 30, 2002, 9:31:49 PM4/30/02
to
In article <3CCF30DC...@halfaya.org> John Leo <l...@halfaya.org> writes:
> and against the the theory that HIV causes AIDS (see
> http://www.virusmyth.net/aids/data/slquestions.htm

Not so surprising. Many people close-minded to a particular science will
*never* recognise that other sciences exist and give results. When they
go to the other science they just go on intuition, which may be correct,
but in many cases is wrong. When you look at books about crackpots you
will find that many FLT-provers, trisectionists, pi=rational provers,
Goldbach provers or denouncers are amateurs. But there are quite a few
that had a tenure at a university. If I remember right, one of those
was the dean. Most of those with education were physisists or
technologically inclined. The moral is, the method to apply in your
part of science is not necesserily the method applied in another science,
and worse, can not applied in that other science.

What I read in the above article, Lang tries to apply strict mathematical
methods to medicine. I.e. he wants a clear definition of what AIDS is,
and wants to use mathematical methods to this. Well, it just aint so.
There is no clear definition of AIDS because it is not yet enought known.
Medicine does not use mathematical methods (except statistics), but only
shows that something is likely, not really possible in mathematics...

Michael Barr

unread,
May 1, 2002, 2:08:46 PM5/1/02
to
"Dik T. Winter" <Dik.W...@cwi.nl> wrote in message news:<GvEs9...@cwi.nl>...

> In article <3CCF30DC...@halfaya.org> John Leo <l...@halfaya.org> writes:
> > and against the the theory that HIV causes AIDS (see
> > http://www.virusmyth.net/aids/data/slquestions.htm
>
> Not so surprising. Many people close-minded to a particular science will
> *never* recognise that other sciences exist and give results. When they
> go to the other science they just go on intuition, which may be correct,
> but in many cases is wrong. When you look at books about crackpots you
> will find that many FLT-provers, trisectionists, pi=rational provers,
> Goldbach provers or denouncers are amateurs. But there are quite a few
> that had a tenure at a university. If I remember right, one of those
> was the dean. Most of those with education were physisists or
> technologically inclined. The moral is, the method to apply in your
> part of science is not necesserily the method applied in another science,
> and worse, can not applied in that other science.
>
> What I read in the above article, Lang tries to apply strict mathematical
> methods to medicine. I.e. he wants a clear definition of what AIDS is,
> and wants to use mathematical methods to this. Well, it just aint so.
> There is no clear definition of AIDS because it is not yet enought known.
> Medicine does not use mathematical methods (except statistics), but only
> shows that something is likely, not really possible in mathematics...

Trouble is that Lang has gone from, "You cannot furnish a mathematical
proof that HIV causes AIDS" to, "HIV does not cause AIDS", which is
about the silliest error a mathematician can make. In addition to
which he has going around giving talks deriding safe sex. This takes
his position from zany to dangerous. At this point, the evidence that
HIV causes AIDS is about as good as epidemiological evidence can ever
be, short of injecting it in people (Lang might be a good candidate)
and seeing what happens.

One problem is that someone (Koch, IIRC) once laid down a set of
criteria for deciding that an infectious agent causes a disease. They
were not engraved on stone tablets, they were just one man's set of
criterions for causality. But they have taken on almost religious
aspects and when they are not satisfied, some people concluded that
causality has not been verified. But they cannot (ethically) be
satisfied here since there is no animal model for the AIDS and no
human volunteers. I believe that certain AIDS-like animal diseases
have been tracked down to HIV-like viruses, but that does not quite
satisfy the criterion.

I would not describe Lang as a kindly old man, although I have not
actually known him since 1964 when I taught a course from a
pre-publication version of his calculus. When I went to his office to
point out an error, he nearly bit my head off. On the other hand, he
has at least one very definite star in his heavenly crown, a rescue
job on someone, but it would invade privacy to say more. Anyway, I
still like his algebra book, at least the shorter first edition.

Bill Dubuque

unread,
May 3, 2002, 3:17:10 PM5/3/02
to

Below is Richard Scott Pierce's conclusion of his Math Review 33:5416
of the 1965 first edition = 508 pp. (vs. 906 pp. 1993 3rd edition)

"Without first-hand experience, it would be difficult to predict the
success of this book in the classroom. Average students will find it hard
going, and only the best will be able to master the immense amount of
material that is laid out by the author. Some persons will find faults in
the book. There are quite a few typographical slips, and a few careless
errors and omissions. It is the reviewer's opinion that the exposition
could be improved in places.

Nevertheless, this book is a remarkable accomplishment. It covers all of
the basic material of abstract algebra in the short span of 500 pages. The
author has an impressive knack for presenting the important and interesting
ideas of algebra in just the `right' way, and he never gets bogged down in
the dry formalism which pervades some parts of algebra. His book is indeed
a worthy successor to van der Waerden's classical transcription of the
lectures of Artin and Noether."

Reviewed by R. S. Pierce

Nobuo Saito

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May 3, 2002, 10:23:58 PM5/3/02
to
ayatollah potassium <san...@bibimus.edu> wrote in message news:<3CCF0D16...@bibimus.edu>...

Could you or anyone else recommend us more up-to-date and better texts
of algebra, then?

Marvin D. Hernandez

unread,
May 5, 2002, 10:15:12 AM5/5/02
to

"Nobuo Saito" <genki...@hotmail.com> wrote in message
news:c1d437f.02050...@posting.google.com...

When I was a grad student at UC at Santa Cruz, Hungerford was used in the
first semester, Lang's was tried in the second semester, then replaced in
the third semester by "Abstract Algebra" by Dummit - first edition (which I
liked the best of the three).

BTW, here is a quote about Lang's book and Hugerford's book from Dr. Lee
Lady from the University of Hawaii
"I used Hungerford most recently, because in previous years students had
complained vociferously about Lang. But I would never use Hungerford again.
Frankly, I think the book sucks. Hungerford seems to have a talent for
making all the wrong choices. In particular, the choice of defining all the
concepts in terms of rings without identity is alone enough to make the book
unusable, in my opinion. This mistake screws up almost every part of the
book where rings are relevant.".

Personally, I like to have several books in my collection because no one
book is perfect.

Marv

Ioannis

unread,
May 6, 2002, 12:28:31 AM5/6/02
to
Marvin D. Hernandez wrote:
> [...] But I would never use Hungerford again.

> Frankly, I think the book sucks. Hungerford seems to have a talent for
> making all the wrong choices. In particular, the choice of defining all the
> concepts in terms of rings without identity is alone enough to make the book
> unusable, in my opinion. This mistake screws up almost every part of the
> book where rings are relevant.".

I humbly second that. Hungerford's Algebra was probably _the_ worst book
I've ever had the priviledge to read. It was my graduate school's
nightmare.

Too little theory on a per/topic basis, and too many hard problems where
the author hopes to familiarize the reader with some of the deeper
concepts. Some of those problems could have been used as explicit
examples instead.

Shaum's series Group Theory beat it easily on most points.

> Marv

--
Ioannis
http://users.forthnet.gr/ath/jgal/
___________________________________________
Eventually, _everything_ is understandable.

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