Let K be a field & Let K be a subfield of L

1.) Let K be a field. If f(x) is a polynomial in K[x] of positive degree and K[x]/(f) is a field, prove that f(x) is irreducible in K[x]

2.)Let K be a subfield of a field L. Let f(x) and g(x) be polynomials in K[x].

(a) If f(x) is a factor of g(x) in L[x], prove that f(x) is also a factor of g(x) in K[x].

(b) If f(x) and g(x) have a common factor of positive degree in L[x], prove that they also have a common factor of positive degree in K[x].

Any help please! Be a lifesaver!