1.For F = R, C, show that Fn is not a field for n > 1 when equipped with the standard addition defined in class and multiplication defined by (x1,...,xn) (y1,...,yn) = (x1y1,...,xnyn), where xi,yi F for i = 1,...,n.

Hint: With this definition, what is the identity element for multiplication? When does an element of Fn have a multiplicative inverse if n > 1?

2.Let F denote R or C. A function p : F F is called a polynomial over F if there exist a0,a1,...,am F

such that

p(z)=a0+a1z++amzm forallzF.

For convenience, we assume that this expression is always chosen so that the coefficient am of the highest power of z is nonzero if m > 0. Under that convention, the nonnegative integer m is called the degree of p. If m=0 and a0 =0, so that p(z)=0 for al lzF, p is called the zeropolynomial.

The collection P(F) of all polynomials over F is a subset of FF, the vector space of all functions F F. Thus, addition and scalar multiplication of polynomials over F is naturally defined by that on FF, and this makes P(F) into a vector space over F. (You should verify this if it is not clear.) With respect to these operations, determine if each of the following subsets of P(F) is a vector space over F, and supply a proof of your answer.

(a) The set Pm(F) of all polynomials over F of degree less than or equal to m, where m is a fixed nonnegative integer.

(b) The set Pm (F) consisting of all polynomials over F of degree equal to m, together with the zero polynomial.