On Quintic Equations

概要

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We consider the quintic equation of the form z^5-az + 1 = 0 (a ∈ - C).When |a| becomes large,we show that its roots ωκ(1<_ k <_5)approach to {0,±a^1/4,±ia^1/4}(i=√<-1>,a^1/4=a 4-th root of a).As an application,we show that when |a| → ∞ galois resolvents Σ^5_i=1 εκωκ(the εκ are distinct 5-th roots of 1)will make five circles centered at the origin on the complex plane.Similar consideration can be applied to higher equations of type z^m - az+1=0,though the distribution of galois resolvents is too complicated to describe.