An algebraic number is a number that is a root of a polynomial with rational coefficients. Any finite combination of rational numbers that can be combined with the usual four operations +, -, *, /, and rational powers can be shown to be an algebraic number. However, not all algebraic numbers can be so defined. So is it possible to write such an algebraic number? It is certainly possible to define a countable infinite of transcendental numbers but not these?
Defining Algebraic Numbers
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abstract-algebra
roots
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0In some constructive approaches to mathematics, real numbers are defined exactly by procedures that give their decimal expansions on demand, as precise as needed. – 2011-06-23
1 Answers
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It depends on what you mean by write. The solutions of $ x^5 − x − 1=0$ are certainly algebraic numbers but they cannot be expressed with an algebraic formula like you describe. (This example is mentioned in wikipedia.)