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Thread: Quadratic Integers

  1. #1
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    Quadratic Integers

    Hello All,

    I have 2 questions regarding quadratic integers.

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

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

    1b. How can it be proven that if $\displaystyle x^3$+$\displaystyle y^3$=$\displaystyle z^3$ and x,y,z are quadratic integers in Q[Sqrt(-3)], then $\displaystyle \lambda$ (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 $\displaystyle \lamda^3$.

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

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

    1b. How can it be proven that if $\displaystyle x^3$+$\displaystyle y^3$=$\displaystyle z^3$ and x,y,z are quadratic integers in Q[Sqrt(-3)], then $\displaystyle \lambda$ (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 $\displaystyle \lamda^3$.

    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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