Galois theory in a different settingSuppose that instead of wanting to express the roots of a polynomial equation with arithmetic operations and radicals we instead wanted expressed it with arithmetic operations and sin(x) ?What (if anything) does Galois theory say about this situation?

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erlentzed · Accepted Answer

Step 1If you mean, when can the roots be expressed using arithmetical operations and sines of rational multiples of   π: because       e          i      x        =  cos  ⁡  (  x  )  +  i  sin  ⁡  (  x  ) the question then boils down when can the roots be expressed in terms of roots of unity?Step 2By Kronecker-Weber, this is possible whenever the Galois extension is abelian.

Galois theory in a different setting. Suppose that instead of wanting to express the roots of a polynomial equation with arithmetic operations and radicals we instead wanted expressed it with arithmetic operations and sin(x) ?

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