Please show the solutions for all 4 parts!
Problem 1 Let m E Z that is not the square of an integer (ie. mメ0, 1.4.9, ). Let α-Vm (so you have a失Q as mentioned above) (i) Prove the following:Qla aba: a,b Q is a subring of C, Za]a +ba: a, b E Z is a subring of Qla], and the fraction field of Z[a] is Q[a]. (3pts) (ii) Prove that Z[x]/(X2-m) Z[a] and Qx/(x2 mQ[a]. (3pts) i Let n be a positive integer. Let I be the principal ideal of Z[a] generated by n. Prove that the quotient ring Z[al/1 has cardinality n2 by showing that the n2 elements u +va: u, v E 0,,n-1) form a complete set of representatives of cosets of I in Z[a]. (3pts) (iv) Let J be a non-zero ideal of Za]. Prove that the quotient ring Z[al/J is finite. (3pts) Hint: parts (i), (ii), and (iii for the case m1 (i.e. the ring Zi] and the field Qi]) have been explained in the lectures and some tutorial notes. The problems here are just straightforward generalization for a general m (if you are confused by this generality, write your work with specific m first such as m 2, -3… until you understand what’s going on then rewrite your work for a general m). Part (iv) needs a bit of ingenuity: prove that there exists a positive integer n in J and use partii)
Prove that Z[x]/(X2-m) Z[a] and Qx/(x2 mQ[a]. (3pts) i Let n be a positive integer. Let I be the principal ideal of Z[a] generated by n nursing assignment tutor.