COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

390 Hans Zassenhaus A real root calculus 391
that U, V are positive between C(G), C(G+ 1) and that the sign of W is
equal to SIGN (W(R)) between C(G), C(G+ 1). Furthermore Q is of
constant sign =b 0 between C(G), C(G+ 1). Moreover neither U and Tnor
V and T have a root in common.
The equation
T(C) = WC9 V(C) - Q(WYC)>
which holds for C = C(G) as well as for C = C(G+ l), implies that T(C) is
positive unless Q(C) P (C) > 0. Hence (26) holds unless
But by our assumption
Q!C(W'(C(GN =- 0,
Q(C(G+ l))P(C(G+ 1)) =- 0.
P(C(G)) P(C(G+ I)) < 0, Q(C(G)) Q(C(G+ 1)) =- 0,
so that (28) cannot hold.
In the general case we have in F[XJ:
where SIGN (U) = 1 and 7? = + GCD (V, T). Furthermore
A
where SIGN (I’) = 1 and ‘F = + GCD( ?, T). Hence
6?= ;P+f’, GCD (6, f’) = 1 = GCD (?, +)a
As was shown above, we have SIGN (?) = 1. Furthermore trivially
SIGN(e) = SIGN (6) = I,
SIGN (7’) = SIGN (a$?) = SIGN (5) SIGN (?) SIGN (?) = 1.
In order to show (22) let us assume P in the form
P(X) = M(0)XrPI+M(l)XIPI-l+ . . . +M([P])
with coefficients in F. Now the equation (20) for
U(X) = M(O)X~~J-1+M(l)X~~-2+ . . . +M([P]-l),
V(X) = X, Q(X) = 1, T(X) = -M([P])
shows that
U(R)V(R) = T(R)
which is tantamount to (22) for the special choice of U, V, T made above.
We have to remark, of course, that for any polynomial U of P[X’J of
degree less than [P] the symbol U(R) is equal to that symbol which is
obtained by substitution of Z(R) in U.
Using the same notations as before, assume that P is separable.
(28)
Denote by A4 the number of sign changes in the chain (8).
As a consequence of the between value theorem there will be constructed
an ordered extension of F of complexity M in which P has M distinct
roots in [A, B). Hence NR(P, A, B) 3 M. On the other hand, let E be an
ordered extension of F with NR(P, A, B) distinct roots in [A, B), say the
roots
M(1) < M(2) < . . . -= M(NR(P, A, B))
by order of magnitude when
A =s M(l), M(NR(P, A, B)) -= B.
There is an ordered extension Z? of E with NR(P’, A, B) roots of P’ in [A, B).
These roots of P’ together with A, B form the chain (8). Since P is separable,
no root of P is a root of P’ and vice versa. Hence each root M(Z) > A lies
between two consecutive members of (8) with a sign change of P between
them, as follows from the mean value theorem. If M(1) = A then by
Rolle’s theorem M(1) < R(J) < M(2) and a sign change of P from A to
R(J) is scored. Therefore there are at least NR(P, A, B) sign changes in
(8). Hence M * NR(P, A, B). Thus Theorem 3 is established.
We remark that for each non-constant polynomial P the polynomial
P/GCD(P, P’) is separable and shares its roots with P. Applying Theorem 3
to this polynomial we obtain Theorem 2.
Theorem 4 is also implied,
In order to prove Theorem 5 let P/GCD (P, P’), and let E be an ordered
extension of F on the level D- 1 of complexity NR(Z’) which is generated
by the adjunction of the NR(2’) roots R(l, 2’) 4 R(2, 2’) -= . . . -C
R(NR(Z’), 2’) of 2’ ordered by magnitude. Let R(0, 2’) = - 00,
R(NR(Z’)+ 1, 2’) = 00, 1 < Z =z NR(P). There is precisely one index J
such that the number of sign changes of P on the subchain R(0, 2’) . . .
R(J, Z’) is equal to I- 1 and that P changes sign from
A = R(J, 2’) to B = R(J+l, Z’).
Using these particular values of A, B we repeat the construction performed
above in order to prove the between value theorem. We use the root symbol
R (I, P) in place of R. We set
SIGN (U, I, P) = SIGN (U(R(I, P))).
In this way we construct indeed an ordered extension F(R(I, P)) of F
which is generated by the adjunction of one root R(I, P) of P subject to the
condition (12) as envisaged in the introduction.
Another application of the induction hypothesis now will yield the proof
of Theorem 5. This is because the definition of the function SIGN (U(R))
in the proof of the between value theorem was forced upon us by the aim
of the construction.
This completes the proof of the string of Theorems l-7 by induction
over the degree of P.
CFA 26