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What was the necessary and sufficient condition that Galois determined for a polynomial?

  • For it to be solvable by radicals

  • For it to be a linear function

  • For it to be a quadratic function

  • For it to be an exponential function

Answer

Galois' necessary and sufficient condition for a polynomial to be solvable by radicals states that a polynomial is solvable by radicals if and only if its Galois group is solvable. In simpler terms, a polynomial can be solved using a finite sequence of algebraic operations (addition, subtraction, multiplication, division, and taking roots) if and only if its group of symmetries (the Galois group) has a specific structure.
Gallant Galois: The Life and Legacy of a French Mathematician

Gallant Galois: The Life and Legacy of a French Mathematician

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