This polynomial is reducible over the real numbers. For example, is a root of this polynomial in . However, it is irreducible over because it has no zeros in . In general if is a any polynomial with real coefficients that hasodddegree then the polynomial is reducible over unless it is linear, try to prove that!

Yes!2. Q the field of rationals, is

A. Q(sqrt(2)) = {a + b(sqrt(2)) | a,b in Q } a subfield of the real numbers?

Remember that is the smallest field containing and . Furthermore, is a vector space with basis and so this field consists of elements of the form .

Yes!B. Q(cube root(3)) = {a + b(cube root(3)) + c(cube root (9)) | a,b,c in Q} a field?

To show that is linearly independent it is is required to show that every finite subset is linearly independent. Let be a finite subset where . If is is linearly dependent then it means for all and not all . But if that is the case then it means that we have a polynomial of degree has only zero values. And so the polynomial must be a zero polynomial. This is a contradiction.3. V_R is the set of all differentiable real valued functions of a real variable, is {1, x, x^2,...} a linearly independent subset of V_R? is it a basis of V_R? Prove your assertions.

It is not a basis. Hint: consider .