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Let $p$ be an odd natural number $\ge 3$. Inspired by results from Euclid's{\em Elements}, we express the irrational $$y=\sqrt[p]{d+\sqrt R}, $$ whosedegree is $2p$, as a polynomial function of irrationals of degrees $\le p$. Incertain cases $y$ is expressed by simple radicals. This reduction of the degreeexhibits remarkably regular patterns of the polynomials involved. The proof isbased on hypergeometric summation, in particular, on Zeilberger's algorithm.

Reducing radicals in the spirit of Euclid