COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

384 Hans Zassenhaus A real root calculus 385
The number of distinct roots of the non-zero polynomial P of F[X] in
any extension of Fis bounded by the degree of P. Hence there is a maximum
NR(P) to the number of distinct roots of P in any ordered extensiont of F.
Similarly, for any two elements A, B of F U {a, - -} satisfying the inequality
A -= B there is a maximum NR(P, A, B) to the number of distinct roots
of P in the interval [A, B) = {YI Y E F & A G Y< B} in any algebraically
ordered extension of F. We have
NR(P) = NR(P, - m, -),
NR(P, A, B) = NR(P, A, C)+NR(P, C, B)
if A -= C -= B,
GW
NR(P, - 03, R(Z, P)) = I- 1, QC)
provided that R(1, P), . . . , R(NR(P), P) are NR(P) distinct roots of P in F
ordered by their order of magnitude. We denote by P’ the derivative
P’(X) = NA(O)Xn--I+@‘- 1)A(l)XIY-2+ . . . + A(N- 1) (3)
of the polynomial
of degree
P(X) = A(0)Xh’+/4(1)X~-l+ . . . $-A(N) (4)
N = [P] (5)
of F[X]. Thus P’(X) = 0 if [P] = 0 01 if P = 0.
We denote by GCD (P, Q) the greatest common divisor with leading
coefficient 1 of the two polynomials P, Q of F[X], not both of which vanish.
There is a well-known routine for finding GCD (P, Q).
The non-zero polynomial P of F[X] is said to be separable if it is not divisible
by any non-constant polynomial square. A necessary and sufficient
condition is given by GCD (P, P’) = 1. In any event, the polynomial P and
the polynomial quotient P/GCD (P, P’) have the same roots.
THEOREM 2. For the non-zero polynomial P of F[Xl and for elements A, B
of F satisfying A < B, there can be constructed an ordered extension of F
that is generated by NR(P, A, B) distinct roots of P belonging to [A, B).
THEOREM 3. Let A, BE F u {- , -->, let P be a separable polynomial of
F[X] and let F contain NR(P’) distinct roots of P’, say
R(1) -= R(2) -== . . . -= R(NR(P’)) (6)
P’(R(I)) = 0 (1 < Z == NR(P’)). (7)
Then the non-negative integer NR(P, A, B) is equal to the number of
changes of sign in the chain of values
P(A), fWJ>)t . . . > J’(R(Kh P(B) (J 6 K> 03)
t By this we mean of course an extension of F with an algebraic ordering which restricts
to the given algebraic ordering on F.
(2b)
where either the indices J, K are so determined that A l, B==R(K+ 1) if K> = 0, (11)
NR(P, - m, R(I, P)) = I- 1, (12)
hence the index I must be a natural number not greater than NR(P).
‘By UP, of course, we denote the polynomial of cE[X] the coefficients of which are
obtained by applying CT to the corresponding coefficients of P.
$ It will be noted that Sturm’s theorem is not needed for our construction, though of
course it provides a very valuable tool in real algebra (see e.g. 121,151). Theorems 6 and 7,
though interesting in themselves, are placed at the end because they do not enter
the construction, but are used only for the purpose of proving the other five theorems.