Abstract algebra question

Senapboebf
timer Asked: Apr 25th, 2015

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(2) (3) 4 (5) On Factoring. Give all irreducible cubics over Z2. Factor p(x)= x + x + x + x + x² +x+1 completely in Z2[x]. How many monic quadratics are there in Z3[x]? Find all irreducible monic quadratics over Zz. Factor q(x)= x + x - - X? - x completely in Z3[x]. Consider the mod function from Z to Zz, and its natural extension to a homomorphism from Z[x] to Z2[x] that mods out the coefficients, for example 5x goes to 2x. Find the image under this homomorphism of the polynomial h(x)= xº +9x® +21x4 +37x² +53,r² + 29x +15. Do the same as in except now one mods out 2. Use parts and Ⓡ to discuss the possible factorization of h(x). For example, does it have any integer roots? Or is it irreducible etc. Discuss as much as you can. 6 8)
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