9 let p x 1 2 1 1 r eur q r this polynomial is irreducible you do not

Question

9. Let p(x) = 1+2+1²+1³ +r¹ € Q[r]. This polynomial is irreducible (you do not need to
prove this). Consider the field obtained as the quotient K = Q[r]/I, where
I = {fp|ƒ €Q[r]}.
Let a = 1 + 1 € K, which is a root of p(x) in K, and recall that every element of K can be
expressed in the form a+ba+co²+do²³, with a, b, c, d e Q.
(a) (8 points) What is the degree of the extension, [K: Q?
[K: Q =
(b) (8 points) Compute the product of 1+a and 2-30³, and express it in the form a +bx+
co²+ da, with a, b, c, d € Q.
(1 + a)(2-30³) =
(c) (8 points) Since K is a field, and a #0, it follows that a is a unit. Find the multiplicative
inverse ¹ and express it in the form a+ba+ca²+da³, with a, b, c, d e Q.