Let L and M be intermediate fields in the extension K ⊂ F.

(a) [LM: K] is finite if and only if [L : K] and [M: K] are finite.

(b) If [LM: K] is finite, then [L : K] and [M: K] divide [LM: K] and [LM: K] ≤ [L: K][M: K].

(c) If [L : K] and [M: K] are finite and relatively prime, then [LM : K] = [L : K][M : K].

( d) If L and M are algebraic over K, then so is LM.