Recall that a complex algebraic variety V in Cⁿ is defined as the set of all solutions of a system of polynomial equation in n complex variables z₁,...,zₙ (the coordinates of Cⁿ). We denote by I_V the ideal in C [z₁,...,zₙ] of all those polynomials which vanish on V, and by C [V] the quotient ring C [z₁,...,zₙ]/Iᵥ (i.e. the ring polynomial functions restricted to V).
1. Let V, W in Cⁿ be two algebraic varieties, V given by equations fᵢ=0, i=1,...,k, and W given by equations gⱼ=0, j=1,...,l, where fᵢ, gⱼ are elements of the ring of polynomials C [z₁,...,zₙ]. Prove that the union of V and W is given by equations fᵢ gⱼ=0, i=1...k, j=1,...,l, and therefore the union of algebraic varieties is an algebraic variety.
2. A map h: W --> V between two sets induces a map of functions: Any function f: V --> C, when composed with h, becomes a function W --> C. Now, suppose that V and W are complex algebraic varieties (in Cⁿ and Cᵐ respectively). Let h : C [V] --> C[W] be a homomorphism of rings which is the identity map on constant functions. Prove that h is induced (in the above sense) by a certain h: W --> V between the sets (and construct h).
3. Let C --> C³ be the map (a parametric curve) given by formulas x=t², y=t³, z=t⁴
t. Describe the range (which is a complex curve in C³) as an algebraic variety, call it V. Let W be the curve in C² with coordinates (u,v) given by the equation u²=v³.
(a) Find an isomorphism between the rings C [V] and C [W].
(b) Show that polynomials f(t) such that f'(0)=0 (f' stands for the derivative of f) form a subring in the ring C [t] of polynomials in one variable, and prove that this ring is isomorphic to the one in part (a).
4. Describe all maximal ideals in R [x] (where R stands for the field of real numbers).

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