1. Define a map f: C^3 \rightarrow C^3 by f(u,v,w)=(u+v+w,uv+vw+wu,uvw). Show that f is onto
but not one-to-one.
2. Show that the polynomial P(X)=X^4+X^3+X^2+6X+1 is irreducible over Q.
3. Show that the polynomial P(x)=1+x/1+x^2/2!+...+x^p/p! is irreducible
over Q (p >= 2 is prime).
4. k is a field. Is the polynomial X^2+Y^2+Z^2 irreducible in k[X,Y,Z] ? What about
5. Let n be a nonnegative integer and a,b in R. Show that the equation X^n+aX+b has
at most 3 real solutions.
6. Solve the system of equations:
7. Let P be a polynomial with real coefficients such that P >= 0. Show that
Q=sum(P^(k),k=0..n) >= 0.
8. Let P in R[X] such that there are infinitely many pairs of integers (u,v) such that
P(u+3v)+P(5u+7v)=0. Show that P has an integer root.
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