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**Homework Statement**

If a is an algebraic integer with a^3 + a + 1 = 0 and b is an algebraic integer with b^2 + b - 3 = 0, prove that both a + b and ab are algebraic integers.

## Homework Equations

An algebraic number is said to be an algebraic integer if it satisfies an equation of the form x^m + c_{m-1}x^{m-1} + ... + c_0 = 0, where the c's are integers.

## The Attempt at a Solution

Since the algebraic numbers form a field, ab and a + b satisfy some polynomial of the form c_mx^m + ... + c_0, where the c's are integers and m <= 6. The problem here is that c_m may not equal 1. I don't know how to get around this. Any tips?