7 8 Marks Let F Be A Subfield Of The Complex Numbers And Let P Be The Abelian Group Of Polynomials Over F Of Degree 1 (42.9 KiB) Viewed 30 times
Please solve a, b, c and explain steps
7. [8 Marks] Let F be a subfield of the complex numbers, and let P₂ be the abelian group of polynomials over F of degree at most 2: P₂ = {a+bx+cx² | a, b = F}. a) Show that if we specify that x³ = x + 1 there is a unique way to make P₂ into a ring, and that this ring is commutative. b) Calculate the product (a+bx+cr²) (d+er+ fx²) in this ring. c) Give examples showing that, depending on the subfield F we begin with, P₂ is sometimes a field, and sometimes not.
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