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**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

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

Solution

**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

**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

**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

**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

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

Solution

**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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