What does the following subring of the complex numbers look like

In summary, the subring of complex numbers generated by the ideal (x) can be represented as {a(x)/b(x) | a(x), b(x) ϵ C[x], b(x) ≠ 0}, and examples of elements in this subring include polynomials with complex coefficients, while examples of elements not in this subring include polynomials with constant terms or combinations of constant terms and imaginary numbers. Additionally, the restriction that b(x) is not in (x) ensures that the elements in the subring do not contain imaginary numbers or constant terms in the denominator.
  • #1
Metric_Space
98
0

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

 
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  • #2


Please provide an attempt.

(and what is q(x)?)
 
  • #3


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

Not sure where to start...
 
  • #4


I think this subring is generated by the ideal (x)
 
  • #5


It's a polynomial ring with coefficients the ratio of two polynomials, right (in C[x])?
 
  • #6


Metric_Space said:
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?
 
  • #7


micromass said:
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?
 
  • #8


Metric_Space said:
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?
 
  • #9


Not sure what the polynomial would look like if the denominator was 0.
 
  • #10


Metric_Space said:
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.
 
  • #11


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

Hmm...ok, I didn't know that.
 
  • #12


what would happen if the restriction b(x) not in (x) was not there? I'm not sure why that restriction might be there...
 
  • #13


Then you would have the ring

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

What does this ring represent? Can you give me some elements of this ring?
 
  • #14


some elements could be x^3+1/x, x^5+2/x^2, etc...right?
 
  • #15


Yes, and these elements are not in the subring

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

do you see why?
 
  • #16


Because they are in (x)?
 
  • #17


What is in (x)? x^3+1/x and x^5+2/x^2 is not in (x)?
 
  • #18


I mean, because x and x^2 are in (x)
 
  • #19


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

[tex]\{a(x)/b(x)\in \mathbbl{C}(x)~\vert~b(x)\notin (x)\}[/tex]
 
  • #20


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)
 
  • #21


Metric_Space said:
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?
 
  • #22


oh..I guess constants? 1, 5, etc
 
  • #23


Yes, but there are other, more interesting, examples...
 
  • #24


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

(1+i),x+i, etc..?
 
  • #25
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

[tex]a_nx^n+...+a_1x+a_0[/tex]

with [itex]a_0\neq 0[/itex] is good.
 
  • #26
...how does knowing this help?
 
  • #27
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?
 
  • #28
The only difference I can see is things not in the subring don't contain i's, constant terms, or combinations of them
 
  • #29
Metric_Space said:
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 [itex]\frac{i}{x(x+i)}[/itex] is also not in the subring. You're right about the constants though (if the constant is not zero, of course)...
 
  • #30
micromass said:
No because [itex]\frac{i}{x(x+i)}[/itex] 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
 
  • #31
Metric_Space said:
I'm not sure how to describe polynomials of this form

polynomials of what form?
 
  • #32
the things you described like i/(x)*(x+i)
 
  • #33
What about it? It's a quotient of the constant polynomial i and the polynomial x(x+i)...
Thus a(x)=i and b(x)=x(x+i)... I don't really see your confusion here.
 
  • #34
I was trying to figure out a way of writing things not in the subring, other than the way already written in the question
 
  • #35
Well, take some elements in (x). And put them in the denumerator of the fraction. Then you obtain things not in the subring.

You obtain things in the subring, by taking elements in (x) and putting them in the denumerator of the fraction...
 

1. What is a subring of the complex numbers?

A subring of the complex numbers is a subset of the complex numbers that forms a ring under the same operations of addition and multiplication as the complex numbers. This means that the subring must contain the additive identity (0) and be closed under addition and multiplication.

2. How is a subring of the complex numbers different from the complex numbers?

A subring of the complex numbers is a smaller set of numbers that still follows the rules of a ring. This means that it may not contain all the elements of the complex numbers, but it still has the same structure and properties as the complex numbers.

3. What are some examples of subrings of the complex numbers?

Some examples of subrings of the complex numbers include the real numbers (which is a subring of the complex numbers), the rational numbers, and the integers. These subsets of the complex numbers still form rings under addition and multiplication.

4. How do you determine if a subset of the complex numbers is a subring?

To determine if a subset of the complex numbers is a subring, you must check that it contains the additive identity (0), is closed under addition and multiplication, and that the additive and multiplicative inverses of each element are also in the subset. If all of these conditions are met, then the subset is a subring of the complex numbers.

5. What does the structure of a subring of the complex numbers look like?

The structure of a subring of the complex numbers is similar to that of the complex numbers. It has an additive identity (0), is closed under addition and multiplication, and has additive and multiplicative inverses for each element. However, it may not contain all the elements of the complex numbers, making it a smaller and more specific set of numbers.

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