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I am wondering if there exists a name for the field $F$ such that $\mathbb{Q}\subset F\subset \mathbb{A}$, and $F$ contains all the radical elements such as $\sqrt[7]{2}, \sqrt[3]{3-\sqrt[4]{7}}, \sqrt[3]{1+\sqrt{4-\sqrt[5]{2}}}$, but not unsolvable elements such as the roots of $x^5+x+1$ ?

Leon Kim
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    Is this the same question as https://math.stackexchange.com/questions/3240951/name-for-numbers-that-are-solutions-to-equations-that-are-solvable-by-radicals? – mweiss Sep 13 '22 at 02:42
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    See also https://math.stackexchange.com/questions/677057/name-for-numbers-expressible-as-radicals?rq=1 and https://math.stackexchange.com/questions/2590563/algebraic-numbers-expressible-in-terms-of-real-valued-radicals – mweiss Sep 13 '22 at 02:46
  • thanks, I did chase through the resources to the answers to those questions, and I found an explicit answer here https://oeis.org/wiki/Algebraic_numbers and they call it the arithmetic numbers – Leon Kim Sep 13 '22 at 02:57
  • The ring (vs field) is sometimes called the ring of radical integers - see Heine's problem - is every solvable algebraic integer a radical integer? (which is still an open problem). – Bill Dubuque Sep 13 '22 at 03:08
  • The OEIS proposed name "arithmetic numbers" is nonstandard. I've never heard it used before (and it's far too vague to be a good name choice). – Bill Dubuque Sep 13 '22 at 03:13
  • Isn't it the maximal solvable extension of $\mathbb{Q}$? Or is that too naive? – Qiaochu Yuan Sep 13 '22 at 03:39
  • I thought these were called explicit algebraic numbers, at least I've been using this term for many years (e.g. see this 25 April 2000 sci.math post). I recall seeing the name used in some of James Pierpont's publications (around 1895 to 1910), and elsewhere in some late 20th century literature. Around 2004 Brian Conrad told me (email) that he liked the term solvable number, (continued) – Dave L. Renfro Sep 13 '22 at 07:35
  • and so I used real-solvable number in this 26 September 2005 sci.math post when the number can be expressed without the necessity of passing through the complex number field, but I've since realized that real radical number is a fairly standard name for such numbers. – Dave L. Renfro Sep 13 '22 at 07:35
  • Actually, looking into this further, I think it was Conrad who used the term real-solvable -- preprint I cited in that 2005 sci.math post -- although it seems to have changed to solvable in real radicals when published, and I believe back then I wasn't entirely sure what term I wanted to use. In any event, it's not even clear to me whether you want real radicals only -- all your examples are with real radicals, but your context doesn't seem to suggest this restriction. – Dave L. Renfro Sep 13 '22 at 07:51

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