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Polynomials, Fields and Derivatives

Determine whether the polynomials have multiple roots.

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19. Let F be a field and let f(x) =...... The derivative, D(f(x)), of f(x) is defined by
D(f(x)) = ......
where, as usual, ....... (n times). Note that D(f(x)) is again a polynomial with coefficients in F.
The polynomial f(x) is said to have a multiple root if there is some field E containing F and some ci c E such that (x ? a)2 divides f(x) in E[x}. For example, the polynomial ..... as a multiple root and the polynomial ..... has .... as multiple roots. We shall prove in Section 13.5 that a nonconstant polynomial f(x) has a multiple root if and only if f(x) is not relatively prime to its derivative (which can be detected by the Euclidean Algorithm in F[xfl. Use this criterion to determine whether the following polynomials have multiple roots:
(a) .....
(b) .....
(c) ......
(d) Show for any prime p and any a E IF that the polynomial x ? a has a multiple root.

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