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Problem 1
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.
Solution

Problem 2
2. Show that the polynomial P(X)=X^4+X^3+X^2+6X+1 is irreducible over Q.
Solution

Problem 3
3. Show that the polynomial P(x)=1+x/1+x^2/2!+...+x^p/p! is irreducible over Q (p >= 2 is prime).
Solution

Problem 4
4. k is a field. Is the polynomial X^2+Y^2+Z^2 irreducible in k[X,Y,Z] ? What about X^n+Y^n+Z^n ?
Solution

Problem 5
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.
Solution

Problem 6
6. Solve the system of equations:
x+y+z=1
x^2+y^2+z^2=21
1/x+1/y+1/z=1
Solution

Problem 7
7. Let P be a polynomial with real coefficients such that P >= 0. Show that Q=sum(P^(k),k=0..n) >= 0.
Solution

Problem 8
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.
Solution


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