Every algebraic number is a critical value of an integer polynomial

Roots of f in ℂ

Closed form

select a root

g

the critical points of g are the real roots of g′

Visual check

GeoGebra

Could not load the Desmos API (it needs the network). Paste these into desmos.com/calculator:

━━ g ━━ g′/k, — divided so its zeros can be read off, which moves no zero; k = 1 in the expression list gives g′ itself ┃ the selected β ┃ the other roots of g′. Every label is Desmos’s own evaluation, not ours.