189-570A: Higher Algebra Assignment 4

Due: Monday, December 4

1. Let K_{1} ⊂ K_{2} ⊂ K_{3} be a sequence of field extensions. Suppose that K_{3} has finite transcendence degree over K_{1}. Show that the same is then true for K_{2}/K_{1} and for K_{3}/K_{2}, and that

tr.deg(K_{3}/K_{1}) = tr.deg(K_{3}/K_{2}) + tr.deg(K_{2}/K_{1}).

2. Let S be a subring of a commutative ring R. An element a ∈ R is said to be integral over S if it satisfies a monic polynomial with coefficients in S. Show that a is integral over S if and only if the subring S[a] generated by a over S is finitely generated as an S-module.

3. Keeping the notation of the previous exercise, show that the set of elements in R that are integral over S is a subring of R.

4. For each of the following ideals I in R[x, y], describe the set V (I) ∩ R^{2}, and write V (I) as a union of irredicible algebraic sets not contained in one another. Give the corresponding expression for I as an intersection of prime ideals.

a) I = (x^{2} + y^{2} − 1, x^{2} − y^{2});

b) I = ((y − 1)(y − x^{2}));

c) I = (y − 1, y − x^{2});

d) I = (y(y − x^{2}));

e) I = (y, y − x^{2});

f) I = (y^{2} − x^{3}, x^{5} − y^{7})

5. For each of the ideals of question 4, state which ideals are non-reduced, and describe the radical of I for these I.

6. Exercises 2, page 826 of Lang’s Algebra (third edition).

7. Exercise 3, page 826 of Lang’s Algebra (third edition).

8. Exercise 4, page 826 of Lang’s Algebra (third edition).

9. Exercise 5, page 827 of Lang’s Algebra (third edition).