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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3You are looking for an algebraic number generating an extension which is not solvable (http://en.wikipedia.org/wiki/Galois_theory#Solvable_groups_and_solution_by_radicals). This should be true of a "generic" irreducible polynomial of degree $5$ or higher (that is, write down a random irreducible polynomial of degree $5$ or higher, and odds are its roots are not solvable). – 2011-06-23
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0That to me is not writing a number. I guess a larger question is why we cannot write such a number? Maybe we need new symbols instead of the rational roots if that is what is required. – 2011-06-23
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1Yes, that's precisely what Abel-Ruffini says (http://en.wikipedia.org/wiki/Abel%E2%80%93Ruffini_theorem): we need new symbols. So what symbols are you willing to admit? – 2011-06-23
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0Maybe not symbols but how about a decimal expansion like was used to generate the first transcendental number. – 2011-06-23
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0How are you allowing the decimal expansion to be described? Given any polynomial there is an algorithm which will, given $n$, output the first $n$ digits of the decimal expansions of the roots. – 2011-06-23
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0Good point. But there does not seem to be a procedure to define the decimal expansion other than the output of some numerical algorithm to solve for the roots. I was again thinking of the decimal expansion of a transcendental number. I guess this is not possible. – 2011-06-23
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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