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Old 2021-08-02, 09:39   #28
Nick
 
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Dec 2012
The Netherlands

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The short answer is that if you are working with a single polynomial equation of degree
at most 4 over the complex numbers, then you can choose any values you like for
all variables except one, and then find the possible values of the remaining variable
by the historical methods passed down to us by Cardano and Ferrari.

To gain more insight into such problems means delving into Algebraic Geometry,
which is a vast and fascinating area of modern mathematics. It takes time to master,
however: where Differential Geometry relies on Calculus (which everyone knows already),
Algebraic Geometry relies on Commutative Algebra (which is new to most people).
A good introduction is Miles Reid's Undergraduate Algebraic Geometry.
For the number theory point of view, the book Rational Points on Elliptic Curves
is a good place to start (but won't cover quartic equations, of course).
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