# Confusion Over Polynomial Separability Example

In lecture notes I am reading for Abstract Algebra, in the Separability section for Field Theory, it gives the introduction and definition that a separable polynomial is one that has no repeated roots in a splitting field. Then it gives an example of a separable polynomial by saying

Thus if $$f(X) = (X-1)^2(X-3)$$ over $$\mathbb{Q}$$, then $$f$$ is separable, because the irreducible factors $$(X-1)$$ and $$(X-3)$$ do not have repeated roots.

Is this correct? Isn't $$(X-1)^2$$ a repeated root at $$X = 1$$?

• the polynomial has a repeated root, but the irreducible factors do not – J. W. Tanner May 27 '20 at 17:41
• @J.W.Tanner What would an irreducible factor with a repeated root look like – Ryan Shesler May 27 '20 at 17:43

You are correct that $$1$$ is a repeated root.
Some authors use an older definition of separability which is that if $$F$$ is a field and $$f(x)=kp_1(x)p_2(x)\dots p_n(x)$$ is any polynomial $$\in F[X]$$ where the $$p_i$$s are all the monic irreducible factors of $$f$$, then $$f$$ is separable if all $$p_i$$s are separable, i.e. all $$p_i$$s have no repeated roots in their respective splitting fields/ in an algebraic closure of $$F$$ (bear in mind, the $$p_i$$s are taken to be irreducible over $$F$$ specifically).
The more modern definition is that a polynomial $$f\in F[X]$$ is separable if it has no repeated roots in its splitting field/ in an algebraic closure of $$F$$.