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Due October 25. Let R denote the set of complex numbers of the form a +...

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Advance Math

Due October 25. Let R denote the set of complex numbers of theform a + b ? 3i, with a, b ? Z. Define N : R ? Z?0, by N(a + b ?3i) = a 2 + 3b 2 . Prove: (i) R is closed under addition andmultiplication. Conclude R is a ring and also an integral domain.(ii) Prove N(xy) = N(x)N(y), for all x, y ? R. (ii) Prove that 1,?1 are the only units in R.

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