# Classification of fields with only separable finite degree fields extensions

The field $\mathbb{Q}$ satisfies the very nice property that if $F$ is a finite degree field extension of $\mathbb{Q}$, then $F / \mathbb{Q}$ is a separable field extension. That is to say, it is algebraic, and for every $\alpha \in F$, the minimal polynomial $m_{\alpha}(x)$ of $\alpha$ over $\mathbb{Q}$ has $\operatorname{deg}(m_{\alpha})$ distinct roots in a splitting field $L$ of $m_{\alpha}$ over $\mathbb{Q}$. This property can be extended to the theorem:

$\textbf{Theorem: }$ Any finite degree field extension of a field of characteristic zero is separable.

My question is does this property extend to any larger class of fields than just those of characteristic zero?

• Such fields are called perfect fields. Finite fields are the simplest examples of perfect fields in positive characteristic. – Brandon Carter May 21 '18 at 21:42
• Thanks Brandon! I checked wikipedia, and it seems it is true that all perfect fields have the desired property. Do we know of any larger class of fields? Or do we know that it is in fact an equivalence? That is to say, if $k$ is a field such that all finite degree field extensions are separable, then is $k$ necessarily perfect? – Adam Higgins May 21 '18 at 21:49
• The property in the OP is an equivalent definition of a perfect field (it is the 3rd equivalent property in the wikipedia link I gave). – Brandon Carter May 21 '18 at 21:50
• @BrandonCarter Oh thanks! I somehow missed that. – Adam Higgins May 21 '18 at 21:51