COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

392 Hans Zassenhaus
It is clear from the construction applied that the set of all polynomials
with coefficients in F of all root symbols obtained by the construction on
all degree levels and of all complexities with the previous operational rules
will yield a constructive definition of an algebraic really closed overfield
of F.
To do the same thing in a more direct manner we observe that algebraic
over algebraic is algebraic so that we establish constructively for any two
root symbols R(I, P), R(J, Q) with PE F[XJ, QC F[x], two further root
symbols R(K, W), R(L, T) (WE F[X], TE F[X]) such that
W, P>+ NJ, Q> = WC W, (2%
W P> WJ, Q) = W, T). (30)
In this way it is shown that the root symbols for polynomials of F[X]
with operational rules (29), (30) and positivity and equality as defined
previously form an algebraic ordered extension P of F. But in P every
polynomial of odd degree has a root constructively as follows at once
from the between value theorem. Similarly, every positive element of P
is a square element. In other words P is really closed.
Again we must emphasize the remark made in A. Hollkott’s thesis that
F is embedded into P only up to isomorphism.
In particular the question whether the root symbol R(Z, P) is equal to
an element of F in standard form R(l, X-A) (A E F) may be effectively
undecidable for ill behaved groundfields (see [l]). However, for F = Q
it is clear that there is an effective procedure for finding all solutions of
P(A) = 0 in Q, the rational number field.
An ALGOL program for the real root calculus over Q has been written
which implements a reduction discovered by H. Kempfert as well as the
Sturm theorem of real algebra (see [5]). It will be discussed in a forthcoming
joint paper by H. Kempfert and the author.
REFERENCES
1. A. FR~HLICH and J. C. SHEPHERDSON: Effective procedures in field theory. Phil.
Trans. Roy. Sot. London A 248 (1956), 401432.
2. AUGUST HOLLKO~: Finite Konstruktion geordneter algebra&her Erweiterungen
von geordneten Grundkerpern. Dissertation, Hamburg, 1941, pp. l-65.
3. ALFRED TARSKI: A Decision Method for Algebra and Geometry. (The Rand Corporation,
Santa Monica, Calif., 1948, III+60 pp.)
4. H. S. VANDIVER: On the ordering of real algebraic numbers by constructive methods.
Ann. of Math. 37 (1936), 7-16.
5. B. L. VAN DER WAERDEN: Algebra, 2,7. Auf]., Heidelberger Taschenbiicher 23.
Some computational problems and methods related
to invariant factors and control theory’
R. E. KALMAN
1. Introduction. The purpose of this modest talk is to point out some
computational problems related to invariant factors in linear algebra. Our
comments are intended as an interim progress report; full details will be
published elsewhere, later.
As is well known, the determination of many invariants in linear algebra
(for instance: minimal polynomials of a vector or a matrix, invariant
subspaces, the rational canonical form of a matrix, the number of eigenvalues
of a matrix in a half-plane or a circle) requires computation in the
polynomial rings R[z] or C[z]. These computations are rather awkward:
first, because they involve checks of divisibility which must be exact;
second, because polynomial arithmetic (especially matrix-valued polynomial
arithmetic) is very awkward to program. Since the numbers desired
are often integers (for instance: the degree of the minimal polynomial of a
matrix), these problems tend to have some of the flavor of finite algebra,
even though strictly speaking they belong to linear algebra.
The question arises: Is it possible to bypass the machinery of polynomial
algebra and relate everything to standard matrix computations, such as the
determination of rank ? This question is of some interest from the viewpoint
of pure mathematics, since it concerns the representation of polynomial
algebra (in the sense analogous to group representations) via matrices.
Even more interesting perhaps are the implications on numerical analysis
and computing art in general, since very little is known today about the
relative numerical advantages and disadvantages of alternate computing
procedures which are abstractly equivalent.
A very interesting and early
“functor” : polynomials + matrice& (1)
is that found in Hermite’s famous paper [l] of 1856 (which introduced
t This work wassupported in part by NASA Grant NgR 05-020-073.
2 We use the term “functor” in a nontechnical sense to mean vaguely: replace some
mathematical object by a (linear) algebraic object.
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