By Andrew O Lindstrum

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**Example text**

The highest-degree term in f¯(x)¯ g(x) must be ≡ 0 (mod p) on the one hand, on the other hand however it is equal to the product of the highest terms of f¯(x) and g¯(x). Since p is a prime and all terms of f¯(x) and g¯(x) are not divisible by p, such product of such terms is also not divisible by p. Consequently the hypothesis is false, and the theorem is proved. ✷ A rational integral polynomial is called primitive if its coefﬁcients are relatively prime. 47 (Theorem of Gauss). The product of two primitive polynomials is again a primitive polynomial.

See G. J. 8 in Chapter I. 7 Norm and trace Let K be a ﬁnite ﬁeld extension of a ﬁeld κ. Take α ∈ K. Then α induces a κ-linear mapping Aα : K −→ K deﬁned by Aα (x) = αx. Let {w1 , . . , wn } be a base of K over κ. Write n Aα (wi ) = αwi = aij wj . j=1 48 1 Field extensions The characteristic polynomial χα (x) = det(xI − Aα ) of the matrix form Aα = (aij ) of Aα is called the ﬁeld polynomial of α. The ﬁeld polynomial χα is independent of the base {w1 , . . , wn } selected for K over κ. Obviously, α is a root of its ﬁeld polynomial.

The ﬁeld E can be given by taking the intersection of all normal extensions of κ containing K. 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.