Qualifying Exam in Algebra

Fall 1995

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1. Let be a finite group, a prime number and a -Sylow subgroup of . Let be the normalizer of in , in other words the subgroup of all such that . Show that is its own normalizer, i.e. that if then . (First show that is the unique -Sylow subgroup of .)
   
   
2. Let be a field and let , be two relatively prime polynomials. Show that is irreducible in .
   
   
3. Let be a polynomial of degree . Let be the splitting field of , and suppose that the Galois group of over is , the symmetric group on letters.
   1. Show that is irreducible.
   2. If is a root of , show that the only automorphism of the field is the identity.
   3. For , show that, with as in b), is not an element of .
   
   
   
4. Let be a ring. Define: is Noetherian. If is Noetherian and is a surjective homomorphism, show that is an isomorphism.
   
   
5. Let be a commutative ring with a multiplicative identity 1. Let be a subset of closed under multiplication and containing 1. If is an -module, define the localization and show that is a functor from the category of -modules to the category of -modules. Prove that localization is exact, in the sense that if
   

   is an exact sequence of -modules, then

   

   is an exact sequence of -modules.
   
   

6. Let , be commutative rings with a multiplicative identity 1.
   1. Let , be commutative rings with a multiplicative identity 1, and suppose that is a ring homomorphism such that . Define what it means for to be integral over .
   2. In the above situation, if and are integral domains and is an inclusion, show that is a field if and only if is a field.
   3. Let be a field. State but do not prove the Noether normalization lemma.
   4. Using (ii) and (iii), prove Hilbert's Nullstellensatz, in the form that if is an algebraically closed field and is a maximal ideal in the polynomial ring , then there exist such that .