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Math Help - Ring in the field of complex numbers

  1. #1
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    Ring in the field of complex numbers

    Hello,

    let k be a field. On the set

    k \left[ i \right] := \lbrace\left(x,y\right) \mid x,y \in k\rbrace

    is defined the addition

    \left( x,y \right) + \left( x',y' \right) = \left(x+x', y+y' \right) ~ \forall x,x',y,y'

    and the multiplication

    \left( x,y \right) \cdot \left( x',y' \right) = \left( xx' -yy', xy' + x'y \right) ~ \forall x,x',y,y' \in k

    Proof that k[i] is a ring with the relations above.

    I have a approach but I am not sure.

    For proofing it, I guess I have to show that it is a commutative Group, that the multiplication is associative und the distributive laws.

    So I want to start with the first. Showing that it is a Abel's group.

    For the group I have to proof the existence of neutral elements, the existence of one inverse element, the associative law and the inner composition "+" must be commutative.

    Okay, so far but how do I show this concretely?

    thanks
    all the best
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  2. #2
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    Quote Originally Posted by Herbststurm View Post
    Okay, so far but how do I show this concretely?

    thanks
    all the best
    Look at the definitions that it needs to satisfy to be a ring.
    And now see if they work out.

    Post your work here so that we can correct it if you get unsure of what to do.
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