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mobius70
Joined: 24 Jan 2008
Posts: 19
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| Posted: Mon Feb 18, 2008 11:03 am Post subject: GR9768 Question # 57 |
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Let R be field of real numbers and R[x] the ring of polynomials in x with coefficients in R. Which of the following subsets of R[x] is a subring of R?
I) All polynomials whose coefficient of x is zero.
II) All polynomials whose degree is even integer, together with zero polynomial.
III) All polynomials whose coefficient are rational numbers.
THE ANSWER SAYS ONLY I AND III ARE CORRECT.
I DON'T GET WHY II IS NOT CORRECT.
--ITS COMPLETE IN ADDITION
--ITS ASSOCIATIVE
--IT HAS IDENTITY ELEMENT ZERO
--FOR EVERY POLYNOMIAL THERE IS AN ADDITIVE INVERSE WHICH WILL ALSO HAVE EVEN DEGREE
--ALSO ADDITION IS ABLIAN
--ITS COMPLETE IN MULTIPLICATION
--ITS ASSOCIATIVEI IN MULTIPLICATION |
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lime
Joined: 04 Dec 2007
Posts: 46
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| Posted: Tue Feb 19, 2008 6:57 pm Post subject: |
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Make it easy! 
| Quote: | --ITS ASSOCIATIVE
--IT HAS IDENTITY ELEMENT ZERO
--FOR EVERY POLYNOMIAL THERE IS AN ADDITIVE INVERSE WHICH WILL ALSO HAVE EVEN DEGREE
--ALSO ADDITION IS ABLIAN
--ITS COMPLETE IN MULTIPLICATION
--ITS ASSOCIATIVE IN MULTIPLICATION |
Those are all correct.
| Quote: | | --ITS COMPLETE IN ADDITION |
That is wrong!! It is not closed under addition:
Two polynomials x^2 and -x^2+x are both of even degree = 2. But their sum
(x^2) + (-x^2 + x) = x
is of degree 1.
Last edited by lime on Wed Mar 12, 2008 7:44 pm; edited 2 times in total |
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mobius70
Joined: 24 Jan 2008
Posts: 19
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| Posted: Wed Feb 20, 2008 5:19 am Post subject: |
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Thx lime ..
You seem to be good here .. :->
just out of curosity ..are you going for some GRE or soemthing |
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