Monday, 16 September 2013

abstract algebra - Multiplicative inverse of a complex number.



Find the multiplicative inverse of 1+32 in the ring Q(2) and use it to solve the equation (1+32)x=152.




I think that the inverse is the conjugate, so it would be 132, but then I don't know where to use in the equation that needs to be solved.


Answer



Let a+b2Q(2) be the inverse of 1+32, i.e. (a+b2)(1+32)=1. Then
1=a+6b+(3a+b)2,
i.e.
3a+b=0, a+6b=1.
It follows that
a=117, b=317.
Now
(1+32)x=152x=(1+32)1(152),
i.e.

x=117(1+32)(152)=3117+8172.


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