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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.
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.
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?
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?