As is well-known, some algebraic real numbers can be expressed using radicals, and others cannot; for example, every real number that is a solution to a quadratic, cubic or quartic equation can be expressed using radicals, while solutions to higher-degree equations may not be. Is there a standard convention for what to call this distinction? I have heard the phrases "expressible numbers" and "inexpressible numbers" used orally, but never seen either one in print.

Context: Many of my students are familiar with the distinction between rational and irrational numbers, but unaware that there are "types" of irrational numbers. Ultimately I would like to be able to say

$$\mathbb Q \subset C \subset B \subset \mathbb A \subset \mathbb R$$

where $C$ denotes those real numbers that can be constructed using compass and straightedge, $\mathbb A$ denotes the algebraic real numbers, and $B$ describes the set of real numbers that can be expressed using radicals. I am asking for a single-word description of the latter, in the same way that "constructible" is a concise description of the former.

  • $\begingroup$ There is the same question here, with no answer either math.stackexchange.com/questions/677057/…, I'm not sure there is even a name for $B$ other than "algebraic numbers expressible by radicals". $\endgroup$ – zwim May 27 at 0:15
  • $\begingroup$ It is a reasonable question, but has been asked before. See Name for numbers expressible as radicals. If you want to have a one word term for them for pedagogical purpose, it will be incumbent on you to introduce that by definition, so perhaps have a bit of fun and call them EARs (expressible as radicals). $\endgroup$ – hardmath May 27 at 0:16
  • $\begingroup$ See also here for the natural variant in which only real-valued radicals can be taken. $\endgroup$ – hardmath May 27 at 0:46

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.