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

Thread starter lahuxixi
Start date Mar 7, 2013
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Integer Unit

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SUMMARY

The discussion centers on proving that an algebraic integer, denoted as α, is a unit in the ring Z(α) when the constant term a₀ of its minimal polynomial is either +1 or -1. The polynomial is expressed as αⁿ + aₙ₋₁αⁿ⁻¹ + ... + a₁α = -a₀. The key conclusion is that if a₀ equals ±1, then α qualifies as a unit, confirming its invertibility within the ring of algebraic integers.

PREREQUISITES

Understanding of algebraic integers
Familiarity with polynomial equations
Knowledge of ring theory
Basic concepts of units in algebraic structures

NEXT STEPS

Study the properties of algebraic integers in number theory
Explore the concept of units in ring theory
Investigate minimal polynomials and their implications
Learn about the structure of Z(α) for various algebraic integers

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Mathematicians, students of abstract algebra, and anyone interested in number theory and algebraic structures will benefit from this discussion.

Mar 7, 2013

lahuxixi

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

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Mar 7, 2013

Dick

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lahuxixi said:

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?