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Metric_Space
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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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Metric_Space said:I think this subring is generated by the ideal (x)
micromass said:Uuh, why would you think that??
Can you list some examples of elements in your subring and not in your subring?
Metric_Space said:Polynomials with complex coefficents should be in the subring
...things divided by 0 shouldn't be, right?
Metric_Space said:Not sure what the polynomial would look like if the denominator was 0.
micromass said:The denominator can never be zero. b(x) is a polynomial, it can not be zero.
Metric_Space said: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)
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
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)...
Metric_Space said:I'm not sure how to describe polynomials of this form
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.
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.
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.
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.
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.