QuestionQuestion

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).

Solution PreviewSolution Preview

This material may consist of step-by-step explanations on how to solve a problem or examples of proper writing, including the use of citations, references, bibliographies, and formatting. This material is made available for the sole purpose of studying and learning - misuse is strictly forbidden.

    $20.00 for this solution

    PayPal, G Pay, ApplePay, Amazon Pay, and all major credit cards accepted.

    Find A Tutor

    View available Abstract Algebra Tutors

    Get College Homework Help.

    Are you sure you don't want to upload any files?

    Fast tutor response requires as much info as possible.

    Decision:
    Upload a file
    Continue without uploading

    SUBMIT YOUR HOMEWORK
    We couldn't find that subject.
    Please select the best match from the list below.

    We'll send you an email right away. If it's not in your inbox, check your spam folder.

    • 1
    • 2
    • 3
    Live Chats