(Re-edited) Let $k$ be a field and $a,b$ algebraic over $k$ but not inside $k$ and $a \neq b$. Suppose that $[k(a,b) : k]=[k(a):k] [k(b):k]$. What does this tell us about the relation between $a,b$?
By the multiplicativity of the degree, we must have $[k(a)(b):k(a)]=[k(b):k]$ and $[k(a)(b):k(b)]=[k(a):k]$.
Can we say anything about the inseparability of e.g. the extension $k \subseteq k(a)$? Can we deduce any other interesting fact?
Edited: Conversely, if $k \subseteq k(a),k \subseteq k(b)$ are algebraic and purely inseparable, can we conclude that $[k(a)(b):k(a)]=[k(b):k]$ and $[k(a)(b):k(b)]=[k(a):k]$?