modulo division with negative power ?

[smslca]

if f(x)modg(x) is valid(means , if it yield a remainder) then , can there be negative powers of x in f(x)?

for example
is (x^-29)mod(x² - 3) possible ?
can we do modulo division like this or is it strictly defined only for positive powers of x?

[R.D. Silverman]

Originally Posted by smslca

if f(x)modg(x) is valid(means , if it yield a remainder) then , can there be negative powers of x in f(x)?

for example
is (x^-29)mod(x² - 3) possible ?
can we do modulo division like this or is it strictly defined only for positive powers of x?

x^(-29) is not a polynomial! This is stuff that should have been covered in high school.
It's a reflection of the sad state of secondary education.

In answer to your question, you first need to define the DOMAIN over
which you are working. Then you need to define what you mean by
x^(-29). It is not a polynomial. Normally when you talk about f(x) mod g(x),
f and g are polynomials are they not?? The fact that x^(-29) is not
a polynomial should have been covered in secondary school.

Your question can be answered, but the answer depends on some stuff
that is not normally covered in high school. Some honors pre-calc
classes might cover it, but I would expect this to be rare.

The answer depends on the RING over which you are working. Discussion
of rings is most often a college level topic. Any good book on modern
algebra will cover it. I can recommend some if you like.

[fivemack]

Originally Posted by smslca

if f(x)modg(x) is valid(means , if it yield a remainder) then , can there be negative powers of x in f(x)?

for example
is (x^-29)mod(x² - 3) possible ?
can we do modulo division like this or is it strictly defined only for positive powers of x?

What would you say ((x^2)/3) was equal to modulo (x^2-3) ?

[R.D. Silverman]

Originally Posted by fivemack

What would you say ((x^2)/3) was equal to modulo (x^2-3) ?

First define 1/3 for the ring in which you are working! One can't answer
a question that has not been properly posed. Posing a question includes
defining the mathematical objects with which one is working.....

[Yes, I know you know this, but others do not]

[davieddy]

How to win friends and influence people.

@Tom. I've found my last but nth "jest" in the usual place. I'm sure you know that no offence was intended. But who moved it?

David

[smslca]

x^(-29) is not a polynomial!

yeah . I read it in wikipedia . Thanks for remembering me

f and g are polynomials are they not??

yes . they are polynomials. which means the above example i have given is not possible.

As i have learned through internet , f mod g is the remainder of the long division of f by g . So cant we continue the division for negative powers and display the remainder involving negative powers.?

The answer depends on the RING over which you are working. Discussion
of rings is most often a college level topic.

I am not a math student . As i am have to study my subjects, it is not possible to study math everyday. It takes me a lot of time to study math.
But i like mathematics to study now or later.
In the mean time can you briefly explain me what the rings are and how they are connected to my example or problem? or suggest any short material on the topic.

[R.D. Silverman]

Originally Posted by smslca

I am not a math student . As i am have to study my subjects, it is not possible to study math everyday. It takes me a lot of time to study math.

Let me give you a little secret: It takes EVERYONE (including math professionals) a lot of time as well.

But i like mathematics to study now or later.
In the mean time can you briefly explain me what the rings are and how they are connected to my example or problem? or suggest any short material on the topic.

There isn't any. One can't learn math at this level "on the cheap". It takes
extensive study to learn it. Furthermore, this medium is not a good setting
for lecturing about math. It is not a blackboard and writing TeX is time
consuming.

However, I can give a quick (and imprecise) definition of a ring: A ring is an "almost" field
where not all of the elements are invertible. i.e. a ring
is "similar" to a field in its arithmetic except for the fact that not elements
have inverses.

A rigorous definition would involve stuff you don't know about. e.g. do
you know what an integral domain is?? Do you understand the concepts
of "characteristic" and "zero-divisor"? A lot of learning modern algebra
is mastering the definitions. By mastering, I don't mean parroting. I mean
being able to use the definitions in proofs and problem solving.

[davieddy]

I'm getting out of here NOW

[science_man_88]

Originally Posted by R.D. Silverman

and writing TeX is time
consuming.

Strtex() in pari turns stuff into TeX.

[smslca]

Here i am concerned only with the remainder that it generates x may be any value and if we consider values of

then can i do the modulo operation on above example.

[R.D. Silverman]

Originally Posted by smslca

Here i am concerned only with the remainder that it generates x may be any value and if we consider values of

then can i do the modulo operation on above example.

???

(1) g(0) is well defined for g(x) = x^2-3, so why exclude it?
(2) +/- sqrt(3) may not even exist in your domain! This is why I said
that one needed to define the domain in which one is working. If you
are working over the real numbers then the concept of remainder itself
vanishes because over the reals everything is exactly divisible by everything
else (except 0 of course). It is a field. Over the rationals, for example,
3 is exactly divisible by 2.

Allow me to repeat what I said earlier: you need to specify the domain
in which you are working. If you are working over a field, then the
very concept of "remainder" does not apply.

You can't determine what x^(-29) mod (x^2-3) is, until you define
what x^(-29) is within your domain. In a field remainders do not exist
in the sense you imply by your question, and in a ring, the element x
may not have an inverse --> it may be a zero-divisor.