cos(pi/q) obtained by real radicals
Anton
I tried to find cos(2*pi/9) in real radicals. In order to find it, I tried to solve the cubic equation 4x^3-3x-1/2=0. I didn't know that it's a Casus Irreducibilis, so it can't be obtained by real radicals. Also it seems that cos(pi/7) can't be found in real radicals because in order to solve the equation x^7-1=0 you can solve two equations, one of them is a quadratic equation and the other is a qubic equation with 3 real roots, so it can't be obtained by real radicals. Can you help me to find cos(pi/q) with rational q which can be obtained by real radicals (except for the case when q=F*2^r, r - rational, F is a prime Fermat number)?
Dave L. Renfro
A little over a year ago I began writing up a
lengthy essay on this and many related topics.
I don't have time to edit the portion that pertains
to your question right now -- I have to leave for
work, but here are some comments and references that
should prove useful. Part of what follows was an
e-mail to Brian D. Conrad at the University of Michigan,
and so some of the references may be repeated.
However, I will say, quickly, that solutions to
x^n = 1 are real-radical if and only if they are
constructible. See Andrzej W. Mostowski's paper
below (which is on-line) and the web page cited
under Dave Rusin's name.
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I will say that a cubic polynomial belongs to the "irreducible case"
if it has rational coefficients and three distinct real irrational
roots. Paolo Ruffini (1765-1822) gave a proof in 1813 that the each
root of an irreducible case cubic is not real-radical expressible.
However, Ruffini's treatment was not sufficiently precise and complete
enough to be fully accepted at the time, and the first completely
satisfactory proof seems to have been given by Pierre Laurent Wantzel
(1814-1848) in 1843. See the section titled "The 'Irreducible Case'
in Cubics" on pp. 345-346 of Cajori [2].
More sophisticated proofs of this were given by Hölder
[4] (1859-1937), Kneser [6] (1862-1930), and Mollame
[7] (1848-1911) in the early 1890's.
This result can also be found in Cajori [1] (the proof is outlined
in Exercise 8 on pp. 208-209), Hadlock [3] (see the solution on
pp. 306-307 to Exercise 5 from Section 3.7), Jacobson [5] (see the
statement of Exercise 7 on p. 259), Thomas [8] (see Theorem 70.1
and its proof on pp. 112-114), and in Waerden [9] (2'nd half of
Section 8.8: "Questions Concerning a Real Base Field", pp. 189-192).
Mostowski essentially proves that trig. values at rational degree
angles are real-solvable if and only if they are constructible.
The Math. Review states: "This is the well-known theorem that
these numbers are expressible by real radicals exactly when phi(n)
is a power of 2."
Generalizations of the irreducible cubic case giving
rise to solvable real numbers that are not real-solvable
(such as your theorem 8.6) can be found in the following
papers.
Irving Martin Isaacs, "Solution of polynomials by real
radicals", Amer. Math. Monthly 92 (1985), 571-575.
Irving Martin Isaacs and David Petrie Moulton, "Real
fields and repeated radical extensions", J. Algebra 201
(1998), 429-455.
Christian Ulrik Jensen, "A remark on real radical
extensions", Acta Arith. 107 (2003), 373-379.
Alfred Loewy, Über die Reduktion algebraischer Gleichungen
durch Adjunktion insbesondere reeller Radikale", Math.
Zeitschr. 15 (1922), 261-273.
http://www.emis.de/cgi-bin/JFM-item?48.0077.01
Finally, in MR 2002f:12003 I saw this remark:
"... a classical result by Loewy, that an irreducible
polynomial of odd degree over a real number field K has
at most one root lying in a real repeated radical
extension of K and in this case all other roots of
the polynomial will be non-real." [I don't know if this
specific result is proved in his 1922 paper, however.]
[17] Andrzej W. Mostowski, "Un théorème sur les nombres
cos 2pi/k/n", Colloquium Math. 1 (1948), 195-196.
[MR 10,43; Zbl 37.30002]
http://www.emis.de/cgi-bin/Zarchive?an=0037.30002
Mostowski's paper is on-line at
http://matwbn.icm.edu.pl/tresc.php?wyd=8&tom=1
[??] Dave Rusin, "When can cos(p*Pi/q) be expressed with real
radicals?", e-mail to Daniel Grubb, 25 November 1996.
http://www.math.niu.edu/~rusin/known-math/96/cosines.grubb
"Vincenzo Mollame (1848-1912) and Ludwig Otto Hoelder
(1859-1937) prove the impossibility of avoiding intermediate
complex numbers in expressing the three roots of a cubic
when they are all real."
http://www.vimagic.de/hope/1/history_eight.html
http://library.wolfram.com/examples/quintic/timeline.html
[??] Bryden R. Cais and Brian D. Conrad, "Modular curves and
Ramanujan's continued fraction", manuscript (April 2004).
At present, this paper is on-line at
http://www.math.lsa.umich.edu/~bcais/papers.html
Section 8 (pp. 33-37) deals with real-solvable numbers.
[??] Florian Cajori, AN INTRODUCTION TO THE THEORY OF EQUATIONS,
Dover Publications, 1904/1969. [JFM 35.0120.03]
http://www.emis.de/cgi-bin/JFM-item?35.0120.03
Reviewed by James Pierpont in Science (N.S.) 21 #525
(20 January 1905), 101-102.
This book is on-line at
http://www.hti.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABV2146
See the proof outlined in Exercise 8 on pp. 208-209.
[??] Florian Cajori, "Pierre Laurent Wantzel", Bulletin of the
American Mathematical Society 24 (1917-18), 339-347.
[JFM 46.0013.04]
http://www.emis.de/cgi-bin/JFM-item?46.0013.04
[??] Brian D. Conrad, "Radical towers and roots of unity",
University of Michigan Math 594 course handout, 2 pages.
http://www.math.lsa.umich.edu/~bcais/594Page/hout/radreal.ps
[??] Charles Robert Hadlock, FIELD THEORY AND ITS CLASSICAL
PROBLEMS, The Mathematical Association of America, 1978.
[MR 82c:12001; Zbl 408.12021]
http://www.emis.de/cgi-bin/MATH-item?0408.12021
See the solution on pp. 306-307 to Exercise 5 from Section 3.7.
[??] Otto Hölder, "Ueber den Casus irreducibilis bei der
Gleichung dritten Grades", Mathematische Annalen 38 (1891),
307-312. [JFM 23.0099.04]
http://www.emis.de/cgi-bin/JFM-item?23.0099.04
Hölder's paper is on-line at
http://134.76.163.65/agora_docs/27607TABLE_OF_CONTENTS.html
[??] Irving Martin Isaacs, "Solution of polynomials by real radicals",
The American Mathematical Monthly 92 (1985), 571-575.
[MR 87d:12006; Zbl 591.12025]
http://www.emis.de/cgi-bin/MATH-item?an=0591.12025
http://www.math.niu.edu/~rusin/known-math/97/real_radical
[??] Irving Martin Isaacs, ALGEBRA: A GRADUATE COURSE, Brooks/Cole
Publishing Company, 1994. [MR 95k:00003; Zbl 805.00001]
http://www.emis.de/cgi-bin/MATH-item?0805.00001
Chapter 22C (pp. 350-354) deals with real-solvable numbers.
[??] Irving Martin Isaacs and David Petrie Moulton, "Real
fields and repeated radical extensions", Journal of Algebra
201 (1998), 429-455. [MR 99a:12003; Zbl 909.12004]
http://www.emis.de/cgi-bin/MATH-item?0909.12004
This paper is on-line at
http://arxiv.org/abs/math.NT/9702232/
[13] Nathan Jacobson, BASIC ALGEBRA I, W. H. Freeman
and Company, 1974. [I have not seen a copy of the 1985
2'nd edition which the following reviews pertain to.]
[MR 86d:00001; Zbl Zbl 557.16001]
http://www.emis.de/cgi-bin/MATH-item?0557.16001
First edition reviewed by Kenneth Bogart in The American
Mathematical Monthly 92 (1985), 743-745.
Second edition reviewed by Andy R. Magid in The American
Mathematical Monthly 93 (1986), 665-667.
See the statement of Exercise 7 on p. 259 (1974 edition).
[10] Christian Ulrik Jensen, "A remark on real radical extensions",
Acta Arithmetica 107 (2003), 373-379. [MR 2004c:12004]
[11] Christian Ulrik Jensen, "Solvability by real radicals and
Fermat primes", Canadian Mathematical Bulletin 47 (2004),
229-236. [MR 2059417]
[14] Adolf Kneser, "Bemerkungen über den sogenannten casus
irreducibilis bei cubischen Gleichungen", Mathematische
Annalen 41 (1892), 344-348. [JFM 24.0095.02]
http://www.emis.de/cgi-bin/JFM-item?24.0095.02
Kneser's paper is on-line at
http://gdz-srv3.sub.uni-goettingen.de/cache/toc/D35930.html
[??] Wolfgang Krull, "Über reelle Radikalkörper", Mathematische
Zeitschrift 65 (1956), 76-90. [MR 18,461i; Zbl 74.03003]
http://www.emis.de/cgi-bin/Zarchive?an=0074.03003
Krull's paper is on-line at
http://134.76.163.65/agora_docs/159603TABLE_OF_CONTENTS.html
[12] Alfred Loewy, Über die Reduktion algebraischer Gleichungen
durch Adjunktion insbesondere reeller Radikale", Mathematische
Zeitschrift 15 (1922), 261-273. [JFM 48.0077.01]
http://www.emis.de/cgi-bin/JFM-item?48.0077.01
Loewy's paper is on-line at
http://134.76.163.65/agora_docs/9038TABLE_OF_CONTENTS.html
[??] Vincenzo Mollame, "Sul casus irreductibilis dell' equazione
cubica", Napoli Rend. (2) 4 (1890), 167-171. [JFM 22.0112.01]
http://www.emis.de/cgi-bin/JFM-item?22.0112.01
[16] Vincenzo Mollame, "Sulle radici primitive dell' unità
negativa", Napoli Rend. (2) 6 (1892), 179-183. [JFM 24.0182.01]
http://www.emis.de/cgi-bin/JFM-item?24.0182.01
Note: A search of "Mollame" at
http://www.emis.de/projects/EULER/search
gives two other items (1895, 1896) that may
also have some historical significance.
[17] Andrzej W. Mostowski, Un théorème sur les nombres
cos 2pi/k/n, Colloquium Math. 1 (1948), 195-196.
[MR 10,43; Zbl 37.30002]
http://www.emis.de/cgi-bin/Zarchive?an=0037.30002
Mostowski's paper is on-line at
http://matwbn.icm.edu.pl/tresc.php?wyd=8&tom=1
[??] Dave Rusin, "When can cos(p*Pi/q) be expressed with real
radicals?", e-mail to Daniel Grubb, 25 November 1996.
http://www.math.niu.edu/~rusin/known-math/96/cosines.grubb
[15] Joseph Miller Thomas, THEORY OF EQUATIONS, McGraw-Hill, 1938.
[JFM 64.0950.04]
http://www.emis.de/cgi-bin/JFM-item?64.0950.04
Reviewed by Fred A. Lewis in The American Mathematical Monthly
46 (1939), 96-97.
See Theorem 70.1 and its proof on pp. 112-114.
[18] Bartel Leendert van der Waerden, ALGEBRA, Volume 1, 7'th edition,
Frederick Ungar Publishing Company, 1970.
[MR 41 #8187b; Zbl 724.12001; ZDM 2004b.01709]
http://www.emis.de/cgi-bin/MATH-item?0724.12001
http://tinyurl.com/3n3zp
See the 2'nd half of Section 8.8: "Questions Concerning
a Real Base Field", pp. 189-192.
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Dave L. Renfro