As preparation for my upcoming PhD written preliminary exam, the following are some sample questions in Algebra, Representation theory, Category Theory and Homological Algebra from various sources (thanks to this mathstackexchange post):
- Let be the field and algebraically closed field containing . Fix , the polynomial . Let be the field of rational functions in an indeterminate .
Prove is a 2-dimensional vector space over . (Checking vector space condition is easy; Dimension being 2 follows from .)
Let be the extension of obtained by adjoining a root of . Show that is a Galois extension with Galois group isomorphic to . (To check is Galois over , note that for , so contains all the roots of ; A bijection from to is given by where . The picture is that is the function field of an abelian Galois cover of of degree .)
How many fields are there such that other than and ? (By Galois correspondence the intermediate fields correspond to subgroups of order , so there are many.)
- Let be a finite group, a normal subgroup of , be a Sylow -subgroup of .
(Frattini’s argument) Show that where is the normalizer subgroup of in . (Given , since is Sylow -subgroup of , there exists such that .)
Show that if every maximal proper subgroup of is normal in , then so is any Sylow -subgroup of . (If is not normal, then we can let be a maximal proper subgroup containing . Then Frattini’s argument implies that , contradiction.)
Let be a finite group, irreducible characters of , subgroup of , irreducible character of . Show that the integers for which satisfy (By Frobenius reciprocity, ; Since is irreducible, this means that for some other . Thus .)
Let be a matrix. Show that . (The subset of diagonlizable matrix is dense and by continuity argument we are done.)
Consider the fields and . Show that , but . (For the last claim, if , then , which implies is constant.)
Prove that if is a finite-dimensional space over a field , and is a nondegenerate bilinear pairing such that for all , then is even. (Find a subspace of codimension 2 such that the restriction of the alternating bilinear form to it is nondegenerate and finish by induction.)
Find the number of elements of order in a simple group of order . (The number of 7-Sylow subgroups is congruent to 1 mod 7 and divides 24 so it is 8. Each 7-Sylow contains six elements of order 7.)
Use the solvability of groups of order 12 to prove that groups of order are solvable. (The 7-Sylow subgroup is normal and its quotient has order 12.)
If and are objects of a category , explain succinctly (but precisely) what is meant by the product of and . (The important thing is that the existence of projection maps with universal property.)
Let be the category with objects being natural numbers and morphisms being set of matrices and composition given by matrix multiplications. Does product exists in this category? (Yes, it is the sum of two natural numbers. Note that this cateogory is equivalent to that of finite dimensional vector spaces.)
What is the sign of the permutation given by in a group and is an element of order and is the order of ? (The answer is . Simply note that each orbit has size and there are many orbits.)
Suppose that the 2-Sylow subgroups of are cyclic and that has even order. Prove that has a subgroup of index 2. (There is a sign homomorphism from to given by the . The hypothesis together with part (a) shows that the image is not identically 1.)
- Establish the irreducibility over of each of the following polynomials:
(Eisenstein polynomial)
(mod 2 is irreducible)
(has no roots in )
Suppose that is an integral domain (i.e., a commutative entire ring). Suppose that and are non-zero ideals of for which the product is a principal ideal. Show that the ideals and are finitely generated. (Suppose with . Show that is generated by .)
Show that the covariant functor from to taking to is nto representable. (It doesn’t preserve limits.)
Note: If we consider it as a contravariant functor it is representable by .
Let be a normal subgroup of order of a finite -group . Prove that is contained in the center of . (Consider acting by conjugation on and compare orders.)
Let be a finite field, and set . For each , let be the product of monoic irreducible polynomial of degree over . Then we have . (Note that is not divisible by square since its derivative is ; The rest is using theory of finite fields.)
Let be a finite Galois extension. Set and let be a subgroup of . Express the group of field automorphisms as a quotient of a subgroup of . (The answer is . Any automorphism of can be extended to , say givin by , which takes to . By Galois correspondence the latter corresponds to . Thus is an automorphism of iff lies in the normalizer of , and belong to iff acts as identity on .)
Let be a prime number different from , and let be a complex -th root of . Set . Show that is a Galois extension of and determine the degree . When , calculate . (Calculate the number of conjugates for the degree and rewrite the minimal polynomial of to get that of .)
Give a counterexample to show that the image of a functor need not be a subcategory. (The problem is that morphisms in the image may not compose.)
Showing taking the center of a group cannot be made into a functor. (Construct such that the composite is an iso and .)
Show that if in , then the number of isomorphism classes of irreducible representations of over is strictly less than the number of conjugacy classes in . (It suffices to find a class function that is not a linear combination of characters. Note that the linear extension of any irreducible character vanish on the element in the group algebra since .)
Let be a subgroup. Determine when the character is irreducible. (Use Frobenius reciprocity, the answer is iff has index at most 2.)
Let be a field extension. Show that two matrices are similar over iff they are similar over . (Use structure theorem of finitely generated modules over PID.)
Prove that submodules of free modules over PIDs are free. (For finite rank modules, we can induct on the rank; More generally, we can do transfinte induction, for details see this answer.)
Show that there exists a chain homotopy equivalence between a chain complex and (with zero differential) iff is split. (Trial and error works, but the more intuitive way is to observe the spliting breaks .)
Show that equivalence between abelian categories is automatically additive (i.e. preserve biproducts) and exact. (Any equivalence between categories can be upgraded to an adjoint equivalence. Then use the fact that left (resp. right) adjoint is right (resp. left) exact (since it preserves any small colimits (resp. limits)).)
Note: The point is that a category being abelian is a property rather than an additional structure, see this and this for more details.