Prove that an algebraic integer is a unit in Z(a)

lahuxixi · Mar 7, 2013

$Typeset text stating: Let α be an algebraic integer and f(x) = x^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 its minimal polynomial. Let R = Z[α]. Prove that α is a unit in R if and only if a_0 = ±1.$

I'm completely stuck here, can anyone help me?

Dick · Mar 7, 2013

lahuxixi said:

$Typeset statement: Let α be an algebraic integer and f(x) = x^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 its minimal polynomial. Let R = Z[α]. Prove that α is a unit in R if and only if a_0 = ±1.$

I'm completely stuck here, can anyone help me?

You shouldn't be completely stuck. One direction at least is easy. ##\alpha^n+a_{n-1}\alpha^{n-1}+...+a_1 \alpha=-a_0##. Can you show that if ##a_0=\pm 1## then ##\alpha## is a unit?

Prove that an algebraic integer is a unit in Z(a)

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