applications of Galois Theory?
Gerry Myerson
applied anywhere. I know this student is not going to be happy
if I try to tell him about class field theory; he means applications
outside of mathematics. This is not my strong point. Anybody care
to fill me in on applications of Galois Theory outside of mathematics?
Thanks.
--
Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email)
W. Dale Hall
Gerry Myerson wrote:
> I'm teaching Galois Theory, and a student asked me whether it's
> applied anywhere. I know this student is not going to be happy
> if I try to tell him about class field theory; he means applications
> outside of mathematics. This is not my strong point. Anybody care
> to fill me in on applications of Galois Theory outside of mathematics?
>
> Thanks.
>
It serves as a good bludgeon for beating the "Top Number Theorist in the
World Today" about the head and shoulders...
Does anyone outside of mathematics care about the existence of field
extensions, their symmetry properties, and the structure of intermediate
fields corresponding to intermediate subgroups of the big symmetry
group? If I had more smarts, I might try to relate the application of
symmetry groups in other areas of mathematics, and ultimately that of
Emmy Noether's theorem (symmetries of the Hamiltonian <--> conserved
quantities) in physics, but I think that may be stretching credibility
a bit (although to me it seems all part of one big idea).
Just my 2 cents' worth.
Dale.
Jim Nastos
> I'm teaching Galois Theory, and a student asked me whether it's
> applied anywhere. I know this student is not going to be happy
> if I try to tell him about class field theory; he means applications
> outside of mathematics. This is not my strong point. Anybody care
> to fill me in on applications of Galois Theory outside of mathematics?
Any kind of information transfer (CD music, phone communication, outer
space communication, internet or networks) uses the theory of
error-detecting and error-correcting codes. Any course in coding theory
I've seen always covers some required field theory.
Doing a google search for GALOIS THEORY CODING has a first hit of a
syllabus from an electrical engineering course. The first section covered
in the course is titled "Galois Theory."
Now, this uses just the basics of field theory and so might not be
exactly the stuff you're teaching. If yours is a very advanced, say,
graduate level math course, then perhaps it will get too abstract for such
applications. But a deep understanding of Galois Theory can't hurt if one
wants to learn (in depth) the ideas to coding theory.
J
Bart Demoen
I do not know much about Galois Theory, but Galois connections are used
heavily in Abstract Interpretation which is a popular form of program analysis
(in some circles). Google for "Galois Theory" and "abstract interpretation".
The seminal work in this area is from Cousot-Cousot 1977
If you google for "Galois Theory" and Cousot, the first thing tha shows up is
something titled:
Galois connections and computer science applications - ...
Cheers
Bart Demoen
SJR
they are used in EE or something. Applications aren't my thing either ;)
David Harden
> Gerry Myerson wrote:
> > I'm teaching Galois Theory, and a student asked me whether it's
> > applied anywhere. I know this student is not going to be happy
> > if I try to tell him about class field theory; he means applications
> > outside of mathematics. This is not my strong point. Anybody care
> > to fill me in on applications of Galois Theory outside of mathematics?
> >
> > Thanks.
> >
>
> It serves as a good bludgeon for beating the "Top Number Theorist in the
> World Today" about the head and shoulders...
I have heard that number theorists are supposed to be averse to Galois
Theory and do things like hope that there is a group which is never
realized as
Gal(K/Q) for an algebraic number field K. However, I don't understand
this, and I would like good examples. Can you provide me with some?
---- David
William Hale
> I'm teaching Galois Theory, and a student asked me whether it's
> applied anywhere. I know this student is not going to be happy
> if I try to tell him about class field theory; he means applications
> outside of mathematics. This is not my strong point. Anybody care
> to fill me in on applications of Galois Theory outside of mathematics?
I think Galois theory inspired Sophus Lie to develop his theory
on Lie groups and Lie algebras, which are applied in modern physics.
This is not a direct application of Galois theory, but I think
it is relevent to the way ideas are developed.
I did a google search with "lie group" "galois theory" and found
quickly this information:
"Lie had started examining partial differential equations,
hoping that he could find a theory which was analogous to
the Galois theory of equations."
The above quote appears at the url:
http://www-groups.dcs.st-and.ac.uk/%7Ehistory/Mathematicians/Lie.html
-- Bill Hale
Jaco Versfeld
> applied anywhere. I know this student is not going to be happy
> if I try to tell him about class field theory; he means applications
> outside of mathematics. This is not my strong point. Anybody care
> to fill me in on applications of Galois Theory outside of mathematics?
>
> Thanks.
Hi Gerry,
Yes, IMHO Galois theory is quite useful. I am doing my D.Eng in
electronic engineering, and my whole thesis is based on the
application of Galois theory. Galois theory is used a lot in
Error-correcting codes (Reed-Solomon codes are a good example, they
are implemented and used with CD's) and Cryptography. It is quite an
interesting field, with a lot of real-life applications...
Regards,
Jaco
G. A. Edgar
Myerson <ge...@maths.mq.edi.ai.i2u4email> wrote:
> I'm teaching Galois Theory, and a student asked me whether it's
> applied anywhere. I know this student is not going to be happy
> if I try to tell him about class field theory; he means applications
> outside of mathematics. This is not my strong point. Anybody care
> to fill me in on applications of Galois Theory outside of mathematics?
>
> Thanks.
If you go to http://www.comap.com/ and search "Galois Theory" you will
come up with their unit "Error-Correcting Codes I".
--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/
W. Dale Hall
Sorry, but I have never heard anything of the sort. The question of
whether a given group can be realized as the Galois group of some
extension of Q is a well-known open problem; I wouldn't be surprised
if there were people on either side of that issue.
The epithet "Top Number Theorist in the World Today" was meant to
indicate someone known to many of us on sci.math, who doesn't rate
as a number theorist at all (there is considerable doubt as to whether
he even understands what a ring is), but who proclaimed in sci.math
that he was the top number theorist in the world today.
Dale.
Chan-Ho Suh
I'm not exactly sure what you mean. I'm pretty sure Galois Theory is a
big deal for number theorists; I know a number theorist that primarily
works with Galois representations, and my impression is that they are very
important for number theory.
Chan-Ho Suh
I think for these kinds of applications the fields used are finite, so
this makes the Galois theory rather trivial. So the class being taught
may be mostly useless (in the sense of these kinds of applications), but I
would guess even the more advanced Galois theory would be useful in the
sense that it needs to be understood to work on number theory applicable
to cryptography.
A good resource for how some algebra and number theory are used in
applications is:
Manfred Schroeder, _Number Theory in Science and Communication_
I don't recall that this really uses any Galois theory, although finite
fields play a role in this book.
MCKAY john
First: let's distinguish between the mathematics of
finite fields F[q], q = p^n. This is not what is of
interest to me here (although I will not pass up on
the opportunity of advertising the Zech logarithm,
giving a neat way to computer the field operations.)
What concerns me, and, I believe the original questioner,
is applications of the theory of finite degree extensions
of the rationals.
The Galois groups are computable by maple for small degree
from input polynomials in one or two variables.
As for applications, the only one I know of outside what one
might expect in mathematics, is determining in what extension
the value of certain integrals lie. I expect James Davenport
knows more on this.
As for real world practical situations, the problem seems to me
to be that one needs some form of stability in that a small
perturbation of the coefficients does not alter the Galois group.
Thus one might match up results with robot motion.
I wish there were more non-mathematical applications - but I do
not know them. Solvability is too precise a notion for real worlds!
In the real world of groups, any distortion, however small,
of a circle turns an infinite group into a finite one. We need a
theory of "approximate symmetry". It doesn't yet exist.
--
But leave the wise to wrangle, and with me
the quarrel of the universe let be;
and, in some corner of the hubbub couched,
make game of that which makes as much of thee.
Dave Rusin
Jim Nastos <nas...@cs.ualberta.ca> wrote:
> Doing a google search for GALOIS THEORY CODING has a first hit of a
>syllabus from an electrical engineering course. The first section covered
>in the course is titled "Galois Theory."
Your google search is probably also going to show that "GALOIS" is
correlated with "THEORY CODING" because finite fields are sometimes
called "Galois fields" (and denoted GF(q) or whatever). Very little
Galois Theory per se is needed to understand finite fields.
dave
Lee Rudolph
...
>As for applications, the only one I know of outside what one
>might expect in mathematics, is determining in what extension
>the value of certain integrals lie. I expect James Davenport
>knows more on this.
>
>As for real world practical situations, the problem seems to me
>to be that one needs some form of stability in that a small
>perturbation of the coefficients does not alter the Galois group.
>
>Thus one might match up results with robot motion.
>
>I wish there were more non-mathematical applications - but I do
>not know them. Solvability is too precise a notion for real worlds!
>
>In the real world of groups, any distortion, however small,
>of a circle turns an infinite group into a finite one. We need a
>theory of "approximate symmetry". It doesn't yet exist.
I am not a 3-dimensional hyperbolic geometer, but I frequently
speak with several of them. Many of the calculations they do
(with such programs as Jeff Weeks's SNAPPEA and Oliver Goodman's
SNAP) involve finding arithmetic invariants of hyperbolic
3-manifolds, and at least some of those calculations (I think)
involve them in Galois theory. For some years now, Jeff Weeks
has been involved with cosmologists in a project intended to
determine whether the geometry of our universe is hyperbolic,
and, if so, which 3-manifold it is. That's a physical instance
(and, in any case, a mathematical instance) where "approximate
symmetries" turn out to be exact, because of the well-known
rigidity theorems for hyperbolic manifolds in dimensions greater
than 2. If Galois theory is involved in those cosmological
applications, then we may have an example of the sort the original
poster wanted. But cosmology is a bit other-worldly for some.
Lee Rudolph
Jim Nastos
...and this is why I stated that it is very elementary "Field Theory" (I
wouldn't even call the algebra used in coding theory as "galois theory"
over the more appropriate "field theory.") I also pointed out that if his
course was an advanced-level one (meaning any course which is not a first
course in abstract algebra) then this probably isn't very useful... but
if you found the webpage to the course he is teaching, you would see that
the course introduces fields and covers the basic fundamental applications
like unconstructibility proofs. In my first course on field theory, we
covered constructibility proofs and the basics to error-detecting and
-correcting codes, so I stand by my suggestion.
J
Keith Ramsay
| applied anywhere.
When I was in college, I saw a job advertisement from one of
the security agencies (I think it was the CIA) looking for
people knowledgable in Galois theory. I think it has applications
to cryptography.
Keith Ramsay
Timothy Murphy
eg why can't an angle be trisected?
(Maybe someone did, and I missed it.)
--
Timothy Murphy
e-mail: t...@birdsnest.maths.tcd.ie
tel: +353-86-233 6090
s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland
Jim Nastos
> I'm surprised no-one has mentioned ruler-and-compass constructions,
> eg why can't an angle be trisected?
> (Maybe someone did, and I missed it.)
The original poster knew of applications to other areas of mathematics,
and I think this would count as one of them... he was more interested in
"applied" applications to actually accomplish something physical.
J
Shmuel (Seymour J.) Metz
at 03:14 PM, "W. Dale Hall" <wd_...@pacbell.net> said:
>The epithet "Top Number Theorist in the World Today"
But in what world?
--
Shmuel (Seymour J.) Metz, SysProg and JOAT
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