Proposition 1

— Let K be a field; then the following properties are equivalent: (AC) Every non-constant polynomial of K[X] splits in K[X] into a product of polynomials of degree 1 (distinct or not). (AC') Every non-constant polynomial of K[X] has at least one root in K. (AC'') Every irreducible polynomial in K[X] is of degree 1.

(AC) Every algebraic extension of K is of degree 1 (in other words, K is relatively algebraically closed in every extension field of K).

Let us first prove the equivalence of the properties (AC), (AC') and (AC''). Clearly (AC) implies (AC). Since every non-constant polynomial of K[X] is divisible by an irreducible polynomial (IV, p. 13, Prop. 13) and every polynomial of degree 1 in K[X] clearly admits a root in K, we see that (AC) implies (AC'). The condition (AC') implies by induction on n, that every polynomial of degree n in K[X] is a product of n polynomials of degree 1 (IV, p. 14, Prop. 1), hence (AC') implies (AC).

It remains to see that (AC) and (AC) are equivalent. If (AC'') holds, every element of an extension field L of K which is algebraic over K is of degree 1 (V, p. 16, Th. 1), hence belongs to K, which establishes (AC). Conversely, let f be an irreducible polynomial of degree $n \geq 1$ in K[X]; the quotient algebra K[X]/(f) is of degree n over K and is a field, hence an algebraic extension of degree n of K (V, p. 18, Prop. 2). Now it is clear that (AC) implies (AC'').