Perhaps S is the same as the "field of real solvable numbers" (I assume the R in your definition is the real numbers). That is, the intersection of R and the field of solvable numbers as defined below. In the paper http://www.ams.org/journals/tran/1968-130-01/S0002-9947-1968-0219416-8/S0002... <http://www.ams.org/journals/tran/1968-130-01/S0002-9947-1968-0219416-8/S0002-9947-1968-0219416-8.pdf> near the bottom of page 47, you will find the following: We shall call an algebraic number a solvable (a "surd") in case its minimal
polynomial is solvable by radicals; i.e., a is a solvable number in case the Galois group of its minimal polynomial is a solvable group. It is clear that the set of solvable numbers is a subfield of the field of algebraic numbers since our definition is equivalent to: "a is solvable if and only if it results from the number 1 by a finite number of rational operations and extraction of roots ((-)1'2, (-)1'3,...)".
On Sat, May 7, 2016 at 4:34 PM, David Wilson <davidwwilson@comcast.net> wrote:
Consider the set S of numbers that can be gotten from elements of Q by addition, multiplication, and taking integer roots? That is,
a in Q => a in S a, b in S => a + b in S a, b in S => ab in S a in R, k in Z, a^k in S => a in S
I understand S would be a proper subset of the algebraic numbers. Does S have a name? Is S a field?
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