Let F be a field of characteristic # 2. Let aand ß be roots of X2 - ae F(X) andX² − b e F(X), respectively. Show that(a + ß) is a root ofX4-2 (a + b) X² + (a - b)².

Since, α is a root of X2 - a = 0, we have α2 - a = 0. => α2 = a Since, β is a root of…

Answered: Let F be a field of characteristic # 2.…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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If A = (aij) = M3 (K) for a field K then the characteristic polynomial of A has the form p₁(x) = - det(A) + C(A)x − Tr(A)x² + x³ where the coefficient of a can be computed as C(A) = a11@22 + a22a33 + a11933 a12a21a13a31 - a23a32 (a) is it necessarily true that if A is invertible then C(A¯¹) = C(A)? (b) is it necessarily true that if det (A) = Tr(A) = 0 then A³ = -C(A)A?
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Let F be a field of characteristic # 2. Let a and ß be roots of X2 - ae F(X) and X² − b e F(X), respectively. Show that (a + ß) is a root of X4-2 (a + b) X² + (a - b)².