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Math Help - Rings and Fields query

  1. #1
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    Unhappy Rings and Fields query

    Hi,

    Just trying to work on my rings and fields.
    If its not too much trouble could someone explain how to do this for me? Its not so much a solution I'm looking for as the method as I really need to understand it

    Let R={a+b ROOT(-14) e C:a,b e Z}
    where C is complex and Z is the real no's

    (i) Prove that U(R)= {+-1}
    (ii) Show that the elements 3,5, (1+ROOT(-14)) and (1- ROOT(-14))
    are irreducible in R
    (iii) By considering the equatioion 3.5 = (1+ROOT(-14))(1- ROOT(-14)) show that 3 is not prime in R
    (iv) Is R a unique factorisation domain?

    I know there are a few concepts involved but I really am in trouble in this subject and help would be much appreciated
    Thanks
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  2. #2
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    10
    Quote Originally Posted by musicmental85
    Let R={a+b ROOT(-14) e C:a,b e Z}
    where C is complex and Z is the real no's
    (i) Prove that U(R)= {+-1}
    The Euclidean norm of an element is a^2+14b^2 this is equal to one if and only if a=\pm 1 , b=0. Thus, \pm 1 are the only units.

    (ii) Show that the elements 3,5, (1+ROOT(-14)) and (1- ROOT(-14))
    are irreducible in R
    (iii) By considering the equatioion 3.5 = (1+ROOT(-14))(1- ROOT(-14)) show that 3 is not prime in R
    (iv) Is R a unique factorisation domain?
    No, it is not a UFD. Once you have have shown that these elements are irreducible elements the equation 3\cdot 5 = (1+\sqrt{-14})(1-\sqrt{-14}) shows there is a different factorization.
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