Lecture 9
Separable Extensions I
 

9. Separable Extensions

Let $ F$ be a field. We have seen that the discriminant of a polynomial $ f(x) \in F[x]$ vanishes if and only if $ f(x)$ has a repeated root. Calculation of discriminant can be difficult. In this section we discuss an effective criterion in terms of derivatives of polynomials to decide whether certain root of $ f(x)$ is repeated. We will also study fields $ F$ so that no irreducible polynomial in $ F[x]$ has repeated roots.
Let $ E$ be a splitting field of a monic polynomial $ f(x) \in F[x]$ of degree $ n.$ Write the unique factorization of $ f(x)$in $ E[x]$.

$\displaystyle f(x) = (x-r_1)^{e_1}(x-r_2)^{e_2}\cdots (x-r_g)^{e_n}.$
where $ r_1,\ldots,r_g \in E$ are distinct and $ e_1,e_2,\ldots,e_g$ are positive integers.


Definition 9.1  

The numbers $ e_1,e_2,\ldots,e_n$ are called the multiplicities of $ r_1,r_2,\ldots,r_n$ respectively. If $ e_i=1$ for some $ i,$ then $ r_i$ is called a simple root. If $ e_i > 1$ then $ r_i$ is called a multiple root. A polynomial $ f(x)$ with no multiple roots is called a separable polynomial.


Proposition 9.2  

The numbers of roots and their multiplicities are independent of a splitting field chosen for $ f(x)$ over $ F.$


Proof.
Let$ E$ and $ K$ be splitting fields of $ f(x)$ over $ F.$ Then there is an $ F-$isomorphism $ \sigma : E\rightarrow K$. This isomorphism gives rise to an isomorphism

$\displaystyle \phi_\sigma :E[x] \rightarrow K[x],\;\; \varphi_\sigma \left(\sum_ia_ix^i\right) = \sum_i\sigma\left(a_i\right)x^i.$

Let % latex2html id marker 15263 $ f(x) = \prod_{i=1}^g(x-r_i)^{e_i}$ be the unique factorization of $ f(x) \in E[x]$. Then % latex2html id marker 15267 $ \phi_\sigma(f(x)) = \prod_{i=1}^g(x-\sigma(r_i))^{e_i}.$ Since $ K[x]$ is UFD, $ \sigma(r_1),\ldots, \sigma(r_g)$ are the roots of $ \phi_\sigma(f(x))=f(x)$ with multiplicities $ e_1,\ldots,e_g$ in$ K$ respectively.


The derivative criterion for multiple roots

Let $ f(x)=a_0+a_1x +\cdots+a_nx^n \in F[x]$. We can define derivative of $ f(x)$ without appealing to limits. This is preferable since F may not be equipped with a distance function.

The derivative of $ f(x),$ is defined by $ f'(x) := \sum_{i=0}^mia_ix^{i-1}$. It is easy to check that the usual formulas for $ (f(x)\pm g(x))',(f(x)g(x))'$ and $ (f(x)/g(x))'$ where $ g(x)\neq0$ hold for derivatives of polynomials.