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.

Solved: 9. Let p(x) = 1+2+1²+1³ +r¹ € Q[r]. This polynomial is irreducible (yo

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

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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