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Topic: Name and methods for the implicit algebraic numbers and functions?
Replies: 7   Last Post: Sep 9, 2015 4:49 PM

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IV

Posts: 354
Registered: 9/1/11
Name and methods for the implicit algebraic numbers and functions?
Posted: Sep 6, 2015 8:31 AM
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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?

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?
3b) Has this incorrect definition of the class of elementary functions
somewhere any consequences?

Thanks.





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