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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.
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3. Show that the polynomial P(x)=1+x/1+x^2/2!+...+x^p/p! is irreducible over Q (p >= 2 is prime).
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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 ?
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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.
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6. Solve the system of equations:
x+y+z=1
x^2+y^2+z^2=21
1/x+1/y+1/z=1
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7. Let P be a polynomial with real coefficients such that P >= 0. Show that Q=sum(P^(k),k=0..n) >= 0.
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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