Resultant
Number theory

The discriminant of a polynomial, which is a fundamental tool in number theory, is a 0 − 1 (− 1) n (n − 1) / 2 res x (f (x), f ′ (x)) {\displaystyle a_{0}^{-1}(-1)^{n(n-1)/2}\operatorname {res} _{x}(f(x),f'(x))}, where a 0 {\displaystyle a_{0}} is the leading coefficient of f (x) {\displaystyle f(x)} and n {\displaystyle n} its degree.

If α {\displaystyle \alpha } and β {\displaystyle \beta } are algebraic numbers such that P (α) = Q (β) = 0 {\displaystyle P(\alpha)=Q(\beta)=0}, then γ = α + β {\displaystyle \gamma =\alpha +\beta } is a root of the resultant res x (P (x), Q (z − x)), {\displaystyle \operatorname {res} _{x}(P(x),Q(z-x)),} and τ = α β {\displaystyle \tau =\alpha \beta } is a root of res x (P (x), x n Q (z / x)) {\displaystyle \operatorname {res} _{x}(P(x),x^{n}Q(z/x))}, where n {\displaystyle n} is the degree of Q (y) {\displaystyle Q(y)}. Combined with the fact that 1 / β {\displaystyle 1/\beta } is a root of y n Q (1 / y) = 0 {\displaystyle y^{n}Q(1/y)=0}, this shows that the set of algebraic numbers is a field.

Let K (α) {\displaystyle K(\alpha)} be an algebraic field extension generated by an element α, {\displaystyle \alpha,} which has P (x) {\displaystyle P(x)} as minimal polynomial. Every element of β ∈ K (α) {\displaystyle \beta \in K(\alpha)} may be written as β = Q (α), {\displaystyle \beta =Q(\alpha),} where Q {\displaystyle Q} is a polynomial. Then β {\displaystyle \beta } is a root of res x (P (x), z − Q (x)), {\displaystyle \operatorname {res} _{x}(P(x),z-Q(x)),} and this resultant is a power of the minimal polynomial of β. {\displaystyle \beta.}
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