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Rings and Modules Seminar
~ Abstracts ~
© 2008 Thomas G. Kucera

R. Padmanabhan
padman(at)cc(dot)umanitoba(dot)ca

Department of Mathematics
University of Manitoba

Wednesday, February 27, 2008

Algebras characterizing constructions with marked ruler and compass
Abstract:

Last month we proved the following
Theorem: A real number a is constructible by an unmarked ruler and compass if and only if there is a tower of subfields of R,

Q = F0 ≤ F1 ≤ F2 ≤ … ≤ Fk ≤ R
such that a is in Fk and each Fi = Fi-1(√ai) for some ai in Fi-1.


Today we will prove a similar theorem that brings out the algebraic meaning of geometric constructions using a marked ruler along with the traditional compass.
Theorem: A real number a is constructible by a marked ruler and compass if and only if there is a tower of subfields of R,

Q= F0 ≤ F1 ≤ F2 ≤ … ≤ Fk ≤ R
such that a is in Fk and each Fi is obtained from the previous Fi-1 by adjoining a root of a quadratic, a cubic or a quartic polynomial. In particular, an arbitrary angle can be trisected by a marked ruler and a compass, and this was known to Archimedes.

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