Separable polynomial

A polynomial over a field is separable when it has no repeated root in a splitting field. Equivalently, it is coprime to its formal derivative.

Separability of roots of unity

m

, then

X^{m} - 1

m X^{m - 1}

, and a common root would have to be both nonzero and zero.

f

is irreducible over a field of characteristic zero, then

f^{'} \neq = 0

and

de g f^{'} < de g f

. Thus

f

cannot divide

f^{'}

, so irreducibility gives

g cd (f, f^{'}) = 1

. Consequently

f

has no repeated root in any extension field.

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A separable polynomial is a polynomial that does not have repeated roots in its splitting field. More formally, a polynomial \( f(x) \) over a field \( K \) is termed separable if its derivative \( f'(x) \) and \( f(x) \) share no common roots in an algebraic closure of \( K \).