What does the following subring of the complex numbers look like

[Metric_Space]

Homework Statement

What does the following subring of the complex numbers look like:

{a(x)/b(x) | b(x) ϵ C[x], b(x) is not a member of (x)} ?

Homework Equations

The Attempt at a Solution

 

[micromass]

Please provide an attempt.

(and what is q(x)?)

 

[Metric_Space]

typo ...it should be b(x).

Not sure where to start...

 

[Metric_Space]

I think this subring is generated by the ideal (x)

 

[Metric_Space]

It's a polynomial ring with coefficients the ratio of two polynomials, right (in C[x])?

 

[micromass]

I think this subring is generated by the ideal (x)

Uuh, why would you think that??

Can you list some examples of elements in your subring and not in your subring?

 

[Metric_Space]

Uuh, why would you think that??

Can you list some examples of elements in your subring and not in your subring?

Polynomials with complex coefficents should be in the subring ...things divided by 0 shouldn't be, right?

 

[micromass]

Polynomials with complex coefficents should be in the subring

Correct.

...things divided by 0 shouldn't be, right?

Uuh, you can't divide by 0... What do you really mean?

 

[Metric_Space]

Not sure what the polynomial would look like if the denominator was 0.

 

[micromass]

Not sure what the polynomial would look like if the denominator was 0.

The denominator can never be zero. b(x) is a polynomial, it can not be zero.

 

[Metric_Space]

The denominator can never be zero. b(x) is a polynomial, it can not be zero.

Hmm...ok, I didn't know that.

 

[Metric_Space]

what would happen if the restriction b(x) not in (x) was not there? I'm not sure why that restriction might be there...

 

[micromass]

Then you would have the ring

$$\{a(x)/b(x)~\vert a(x), b(x)\in C[x], b\neq 0\}$$

What does this ring represent? Can you give me some elements of this ring?

 

[Metric_Space]

some elements could be x^3+1/x, x^5+2/x^2, etc...right?

 

[micromass]

Yes, and these elements are not in the subring

$$\{a(x)/b(x)~\vert~a(x),b(x)\in C[x],~b(x)\notin (x)\}$$

do you see why?

 

[Metric_Space]

Because they are in (x)?

 

[micromass]

What is in (x)? x^3+1/x and x^5+2/x^2 is not in (x)?

 

[Metric_Space]

I mean, because x and x^2 are in (x)

 

[micromass]

Yes, can you give me an element not in (x)? And can you use this to make an element which is in

$$\{a(x)/b(x)\in \mathbbl{C}(x)~\vert~b(x)\notin (x)\}$$

 

[Metric_Space]

I know (x) is

(x) = xR = {r ϵ R | r = at for some t ϵ R}

Not sure how to find some element not in (x)

 

[micromass]

I know (x) is

(x) = xR = {r ϵ R | r = at for some t ϵ R}

OK, that last equation makes no sense since it doesn't even contain x.

Not sure how to find some element not in (x)

Well, (x) contains all the polynomails that are a multiple of x. For example x, x(x+1), x² are all multiples of x. Can you give me something that is not a multiple of x?

 

[Metric_Space]

oh..I guess constants? 1, 5, etc

 

[micromass]

Yes, but there are other, more interesting, examples...

 

[Metric_Space]

ah...anything with complex #'s...

(1+i),x+i, etc..?

 

[micromass]

Yes 1+i is good. And x+i is also good. But x+1 would also have been good. And x²+4x+1 too.

Basically, any polynomial

$$a_nx^n+...+a_1x+a_0$$

with $a_0\neq 0$ is good.

 

[Metric_Space]

...how does knowing this help?

 

[micromass]

Uuh, well, because now you can actually give examples of elements in your subring. You should never attempt a question before you actually know and understand all the things involved. That includes knowing various examples.

So, can you now give me 3 or 4 examples of things in your subring and things not in your subring? What diffference do you notice?

 

[Metric_Space]

The only difference I can see is things not in the subring don't contain i's, constant terms, or combinations of them

 

[micromass]

The only difference I can see is things not in the subring don't contain i's, constant terms, or combinations of them

No because $\frac{i}{x(x+i)}$ is also not in the subring. You're right about the constants though (if the constant is not zero, of course)...

 

[Metric_Space]

No because $\frac{i}{x(x+i)}$ is also not in the subring. You're right about the constants though (if the constant is not zero, of course)...

I'm not sure how to describe polynomials of this form