By David M Bressoud; S Wagon

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Let κ be a ﬁeld, and 0 = f(x) ∈ κ[x]. If f(x) has no multiple root in an algebraic closure κ of κ, then f is called a separable polynomial. Let K be an extension of κ, and let α ∈ K be algebraic over κ. If the minimal polynomial of α over κ is separable, then α is called a separable algebraic element over κ, otherwise, α is inseparable over κ. If an inseparable element over κ exists, then char(κ) = 0, where char(κ) denotes the characteristic of κ. 68. Let K be an algebraic closure of κ. Take α ∈ K and let Pα be the minimal polynomial of α over κ.

Thus we see that ΩB/A exists. It follows from the deﬁnition that the pair (ΩB/A , d) is unique up to unique isomorphism. As a corollary of this construction, we see that ΩB/A is generated as a B-module by {dx | x ∈ B}. Concretely, we may consider the homomorphism m : B ⊗A B −→ B, x ⊗ y → xy, whose kernel we denote by I. 6) is a B ⊗ B-module, and hence in particular also a B-module, via the embedding B −→ B ⊗ B, x → x ⊗ 1. If we put dx = x ⊗ 1 − 1 ⊗ x mod I 2 , then we obtain an A-derivation d : B −→ ΩB/A .

Xm ) If we substitute for β1 , . . , βm in R their representations as polynomials in α, βi = gi (α) ∈ κ[α], i = 1, 2, . . , m, then Q(β1 , . . , βm ) becomes a polynomial in α, which does not vanish for the value α. Consequently, it does not vanish for any of the conjugates α(1) , . . , α(n) of α with respect to κ. However, P (β1 , . . , βm ) = P (g1 (α), . . , gm (α)) = 0. 40 1 Field extensions Hence this polynomial in α must vanish for all conjugates α(1) , . . , (i) (i) = P g1 α(i) , .