XPost: sci.math
From: hebisch@antispam.uni.wroc.pl
In sci.math.symbolic IV wrote:
> Hallo.
>
> 1)
> We know that some polynomials have solutions, but their Galois groups are
> not solvable. Therefore there are algebraic numbers which cannot be
> explicitly described by an algebraic expression.
> 1a) Is there a name for this kind of algebraic numbers?
> 1b) Is there a method to construct or describe this kind of algebraic
> numbers only by symbolic methods, e.g. by closed-form expressions?
>
> 2)
> We know, the defining algebraic equation of some algebraic functions has a
> solution, but the Galois group of the eqaution is not solvable. Therefore
> there are algebraic functions which cannot be explicitly described by an
> algebraic expression.
> 2a) Is there a name for this kind of algebraic functions?
> 2b) Is there a method to construct or describe this kind of algebraic
> functions only by symbolic methods, e.g. by closed-form expressions?
1 and 2 areessentially the same question. For subpoint b: there
is easy to represent roots of irreducible polynomial P in symbolic
way. Just introduce a new name (say b) for a root and use defining
relation P(b) = 0 for simplification.
For example if P(x) = x^3 + x + 7, then you replace b^3 by
(-b - 7) and similarely for higher powers. Division is a bit
more tricky. It uses extended euclidean algorith. If y = Q(b)
where Q is a polynomial of degree smaller than degree of P, then
we can find polunomials U and V
such that
1 = U(x)P(x) + V(x)Q(x)
Then we have 1 = V(b)Q(b), so V(b) is the inverse of Q(b).
This is implemented in all major computer algebra systems, usually
using name like rootOf(P) or RootOf(P).
Concerning a, AFAIK there are no special names: normally when
talking about algebraic numbers or functions one talks about
roots of polynomials and one ignores problem of expressing
given number or function in terms of radicals.
> 3)
> Liouville and Ritt ("Integration in finite terms") define the class of
> elementary functions by algebraic equations over algebraic or transcendent
> monomials. Considering 1) and 2), this means that their class of elementary
> functions contains also the functions which are not expressible by an
> algebraic expression. This contradicts the normal use of the term
> "elementary function".
> 3a) Am I right?
Partially, but mostly wrong.
> 3b) Has this incorrect definition of the class of elementary functions
> somewhere any consequences?
"elementary function" were defined by Liouville. Since this is
definition you can not call it wrong. In fact, the intent was
to wark with functions given in "finite terms". Roots of
polynomials are described by finite amount of data and one
can do computations with them, so they are given in
"finite terms". At first Liouville worked only with expressions
in terms of radicals. But after reading Abel's work Liouville
realized that roots of polynomials are more general and do not
pose any extra difficulty compared to roots. So he and followers
consider current definition better, then one insisting of
expressing algebraic quantities in terms of radicals.
BTW: Book
J.H. Davenport, Y. Siret, E. Tournier, Computer Algebra -- Systems
and Algorithms for Algebraic Computation
is available at:
http://staff.bath.ac.uk/masjhd/masternew.pdf
In chapter 2 the authors explain representation of functions
in a CAS, in particular representation of algebraic numbers
and functions.
BTW2: We can represent in "finite terms" much more than
elementary functions -- elementary functions are just a
start.
--
Waldek Hebisch
hebisch@antispam.uni.wroc.pl
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