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Math Help - Quadratic Integers

  1. #1
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    Quadratic Integers

    Hello All,

    I have 2 questions regarding quadratic integers.

    1a. Assume that \lambda= (3+Sqrt(-3))/2 exists in Q[Sqrt(3)]. How can we show that if x= 1(mod \lambda)^3 that x^3= 1(mod \lambda)^3? Also, how can we show that if x= -1(mod \lambda), then x^3= -1(mod \lambda)^3? Lastly, if x= 0(mod \lambda), how can we show that x^3= 0(mod \lambda)^3 ?

    I know that \lambda is prime and that there are three congruence classes (mod \lambda): one with -1, 0, and 1.

    1b. How can it be proven that if x^3+ y^3= z^3 and x,y,z are quadratic integers in Q[Sqrt(-3)], then \lambda (as described in 1a) must divide of one x, y, or z?

    My first guess at this one is to reduce the equation to modulo \lamda^3.

    All help is appreciated!
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  2. #2
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    Quote Originally Posted by Samson View Post
    Hello All,

    I have 2 questions regarding quadratic integers.

    1a. Assume that \lambda= (3+Sqrt(-3))/2 exists in Q[Sqrt(3)]. How can we show that if x= 1(mod \lambda)^3 that x^3= 1(mod \lambda)^3? Also, how can we show that if x= -1(mod \lambda), then x^3= -1(mod \lambda)^3? Lastly, if x= 0(mod \lambda), how can we show that x^3= 0(mod \lambda)^3 ?

    I know that \lambda is prime and that there are three congruence classes (mod \lambda): one with -1, 0, and 1.

    1b. How can it be proven that if x^3+ y^3= z^3 and x,y,z are quadratic integers in Q[Sqrt(-3)], then \lambda (as described in 1a) must divide of one x, y, or z?

    My first guess at this one is to reduce the equation to modulo \lamda^3.

    All help is appreciated!
    Does reducing this to modulo 3 do me any good? Can anyone offer some pointers on this?
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