[math-fun] Question about a set
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?
See https://en.wikipedia.org/wiki/Galois_theory The standard example given there of z notin S is the real root z of x^5 - x - 1 = 0 . Any ring generated over rationals |Q by some root z of polynomial P(x) = c_n x^n + ... + c_0 is necessarily a field, since reciprocal (and division) is possible via 1/z = (-c_n/c_0) z^(n-1) + ... + (-c_1/c_0) . The fact that the extension field is the quotient of polynomial ring |Q[x] by (ideal generated by) P(x) seems unfamiliar to a number of otherwise competent computing mathematicians, contributing recently to an excruciating "gotcha" surfacing in a certain strongly-typed CAS which shall remain nameless ... Fred Lunnon On 5/7/16, 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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Omission: below P(x) must be irreducible, so minimal degree for given z ! WFL On 5/7/16, Fred Lunnon <fred.lunnon@gmail.com> wrote:
See https://en.wikipedia.org/wiki/Galois_theory The standard example given there of z notin S is the real root z of x^5 - x - 1 = 0 .
Any ring generated over rationals |Q by some root z of polynomial P(x) = c_n x^n + ... + c_0 is necessarily a field, since reciprocal (and division) is possible via 1/z = (-c_n/c_0) z^(n-1) + ... + (-c_1/c_0) .
The fact that the extension field is the quotient of polynomial ring |Q[x] by (ideal generated by) P(x) seems unfamiliar to a number of otherwise competent computing mathematicians, contributing recently to an excruciating "gotcha" surfacing in a certain strongly-typed CAS which shall remain nameless ...
Fred Lunnon
On 5/7/16, 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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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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participants (3)
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David Wilson -
Fred Lunnon -
W. Edwin Clark