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What is the name (there must be one!) for real (or perhaps complex) numbers expressible as radicals? Radical numbers? Solvable numbers? (following the same logic as ‘solvable group’). In other words, fill in the blank: Abel proved that there are algebraic numbers that are not _ numbers.

Just to be definite, I want a word for those real numbers that result from exactly evaluating an expression built out of only the constants 0 and 1, the binary operations of addition, subtraction, multiplication, and division, and the infinite family of unary operations of square root, cube root, etc; forbidding division by 0 or even-index roots of negative numbers (and taking the positive even-index roots of positive numbers). But if you know something similar for complex numbers, then that would be nice too.

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  • $\begingroup$ The word you're looking for is either “radicals”, or no word at all. Likewise, there's no special name for those algebraics that can not be expressed as such either: except for those that are the roots of fifth order polynomials, and which are also called “Bring radicals”. $\endgroup$ – Lucian Feb 15 '14 at 5:42
  • $\begingroup$ They are called algebraic numbers. As I see it, your construction allows exactly those numbers which are the root of a polynomial with integer coefficients. That's what they are called mathworld.wolfram.com/AlgebraicNumber.html $\endgroup$ – Peter Webb Feb 15 '14 at 6:38
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    $\begingroup$ @PeterWebb: Not all algebraics are expressible in radicals ! $\endgroup$ – Lucian Feb 15 '14 at 19:16
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That are the real algebraic numbers that are defined/expressible by radicals (radical expressions), the explicit real algebraic numbers.

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  • $\begingroup$ Ah, thank you, I see that this term is used in Mathematica documentation and appears as Definition 2.46 of Computer Algebra and Symbolic Computation: Mathematical Methods by Joel S. Cohen. (This is for the complex case, but adding the word ‘real’, as you did, fixes that.) (I also found a paper that defined it in a different way, although since their definition seemed to be equivalent to simply an algebraic number, either they or I didn't understand what they were doing.) $\endgroup$ – Toby Bartels Feb 22 '19 at 23:30
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I am not sure if there is a generic name. However, certain forms of radicals had specific names in Euclid's time. Generally, depending on the use of the radicals, they were called, first binomial, second binomial, etc up to fifth binomial and similarly first through fifth apotome. Sixth was used for those beyond fifth.

See

http://aleph0.clarku.edu/~djoyce/java/elements/bookX/bookX.html#defsII

and

http://aleph0.clarku.edu/~djoyce/java/elements/bookX/bookX.html#defsIII

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