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1 Homework 1 - University of Pennsylvania

1 Homework 1. (1) Prove the ideal (3,x) is a maximal ideal in Z[x]. SOLUTION: Suppose we expand this ideal by including another generator polynomial, P / (3, x). Write P = n + x Q with n an integer not divisible by 3 (if 3|n then P (3, x) so we have not expanded the ideal) and Q is some polynomial. Then subtracting off a multiple of one generator from another does not change the ideal (analogous to row operations on a matrix not changing the row space), so in particular (3, x, P ) = (3, x, n). As n is not a multiple of 3, gcd(3, n) = 1, so 1 (3, x, P ) and thus (3, x, P ) is all of Z[x]. Thus (3,x) is maximal. (2) Prove that (3) and (x) are prime ideals in Z[x]. SOLUTION: If P and Q are polynomials, then the constant term of P Q is the product of the constant terms of P and Q. Thus, if P Q (x) then the product of their constant terms is 0, and since Z is an integral domain, this means one of them has a constant term equal to 0, hence lies in (x).

(5) Let F be a field and F[[z]] the ring of power series. (a) Show that every element A= P ∞ i=0 a iz i such that a 0 6= 0 is a unit in F[[z]]. SOLUTION: We want to show that there is B= P ∞ i=0 b iz i such that AB= 1 + 0z+ 0z2 + ...The coefficient b 0 must therefore be a−1 0 which exists since a 0 is assumed to be a nonzero element in a ...

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