Abel’s Theorem

Harold M. Edwards

Chapter Chapter 9 in Essays in Constructive Mathematics, 2022, pp 265-306 from Springer

Abstract: Abstract This chapter states and proves four Abel’s theorems inspired by his 1826 Paris memoir and last published paper. Abel studied transcendental functions $$\int \! f(x,y)dx$$ ∫ f ( x , y ) d x , where f(x, y) is a rational function and y is related to x via the polynomial equation $$\chi (x,y) = 0$$ χ ( x , y ) = 0 , though his proofs often used the differential $$f(x,y)dx$$ f ( x , y ) d x . The essays in this chapter take a differential point of view in order to recast Abel’s ideas as theorems in constructive algebra. The holomorphic differentials encountered in Chapter 4 have a natural definition using the points constructed in Chapter 8. The final four essays explore different aspects of Abel’s ideas in which holomorphic differentials and the genus play prominent roles. Examples are included to reveal the connection to Euler’s addition formula.

Date: 2022
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-030-98558-5_9

Ordering information: This item can be ordered from
http://www.springer.com/9783030985585

DOI: 10.1007/978-3-030-98558-5_9

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().