5. Show that every polynomial ƒ € F[x] has a nonzero multiple such that every exponent is prime. I.e., 0‡ g = fh such th

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5 Show That Every Polynomial F F X Has A Nonzero Multiple Such That Every Exponent Is Prime I E 0 G Fh Such Th 1 (132.42 KiB) Viewed 12 times

5. Show that every polynomial ƒ € F[x] has a nonzero multiple such that every exponent is prime. I.e., 0‡ g = fh such that g = Σ apap p prime for some coefficients ap. (Hint: there are infinitely many primes). 6. Let Y and Z be subspaces of the finite dimensional vector spaces V and W, respectively. Suppose that T: V → W is a linear map such that T(Y) ≤ Z. Show that T induces a linear map T : V/Y → W/Z defined by T(v + Y) = T(v) + Z. Consider a basis ß = {v₁, ..., Vn} of V containing a basis ß′ = {v₁, ..., Uk} of Y and a basis y {w₁,..., wm} of W containing a basis y' = {w₁, ..., we} of Z. Show that the matrix M(T) representing T is a block matrix of the form = A C (68) Explain how to determine the matrices M (Ty) and M?" (T) from this block matrix, where " = {Uk+1+Y, ..., Vn +Y} and y″ = {we+1+Z,..., Wm+Z} are bases of V/Y and W/Z. B"