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I have been studying the Pisot-Vijayaraghavan numbers recently however I have little background in the basic theory of algebraic numbers. There are a number of standard results which the texts give without proof and which I haven't yet been able to prove myself.

Here are a few that are bugging me:

Let a = a(1) be an algebraic integer with conjugates a(2) , ... , a(k), then a(1), a(2), ... ,a(k) are distict.

Let a = a(1) be an algebraic integer with conjugates a(2) , ... , a(k), then a^n = a(1)^n is an algebraic integer with conjugates a(2)^n , ... , a(k)^n.

Let a = a(1) be an algebraic integer with conjugates a(2) , ... , a(k) and let P(x) be a monic polynomial with rational integer coefficients, then P(a) =/= 0 implies P(a(i)) =/= 0 for all i = 2 , ... , k.

Let a = a(1) be an algebraic integer with conjugates a(2) , ... , a(k) and let P(x) be a monic polynomial with rational integer coefficients, then P(a) is an algebraic integer with conjugates P(a(2)) , ... , P(a(k)).

Any hints on how to prove the above would be very useful.

Regards.

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# Algebraic Number Theory

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