87d:12006 12F05 51M15
Isaacs, I. M.(1-WI)
Solution of polynomials by real radicals. (English)
Amer. Math. Monthly 92 (1985), no. 8, 571--575.
Elementary Galois theory is normally done over complex extensions of
the rationals, since the subject has such a complete and attractive
formulation there. However, in this note, the author proves an
unexpected result about the restricted case of real extensions of the
rationals.
The basic situation remains the same. An extension is radical if it is
obtained by repeated adjunction of roots. In general, even if all the
roots of a polynomial are real, they are not obtainable through the
extraction of real roots. However, in the case when a real extension
is radical, there is a strong restriction on its degree.
Suppose that all the roots of an irreducible polynomial $f(x)$ over
${\bf Q}$ are real. Then if any root of the polynomial is radical, it
is shown that the degree of the polynomial must be a power of two. It
is an easy corollary that, under the conditions given, all the roots
of the polynomial are constructible with straightedge and compass.
This note is written in the author's customary scrupulous style. It
would make a good reading project for a capable undergraduate or
beginning graduate student.
Reviewed by D. W. Erbach
© Copyright American Mathematical Society 1987, 1997