Standard Generals Questions - Algebra Matrices -------- Eigenvalues of a symmetric matrix. Which matrices are diagonalizable? What happens if two diagonalizable matrices commute? What is an orthogonal matrix? What is a Toeplitz operators? For A skew-symmetric, show S=A+I/A-I is orthogonal. Express A in terms of S. Show that the eigenvalues of a skew matrix are imaginary. How do you diagonalize a symmetric matrix geometrically? What 3x3 polynomials over the rationals, up to similarity, satisfy f(A)=0, where f(x)=(x^2+2)(x-1)^3? ... where f(x)=x^4-x^3-x+1? Prove the fact that some combination of skew-symmetric matrix give a orthogonal matrix (sic). What are the eigenvalues of a hermitian matrix? Of a unitary matrix? When do the powers of a matrix tend to zero? How do you know that an operator satisfies its characteristic polynomial? What is the name of this theorem? Vector spaces ------------- What is a bilinear form over a vector space? When are two forms equivalent? What is an orthogonal matrix and what relevance does it have to the above? Galois theory and field theory ------------------------------ Explain Galois theory. Give an example. What are the automorphisms of a finite field? Can you have a nonseparable extension of Q or Z/p? Why not? Give an example of a nonseparable extension. is Q(cbrt(2)) normal? What is its splitting field? What is its lattice of subfields? What is the Galois group of x^(2^n-2), x^2-2, x^5-2, x^4-2, x^2+9, x^n-1? Discuss sufficient conditions on a polynomial of degree 5 to have Galois group S_5 (two nonreal roots.) What are necessary and sufficient conditions for the Galois group to have an order a multiple of 3. Give a condition on the Galois group that is implied by the irreducibility of the polynomial. What does square discriminant imply about the Galois group of a polynomial? What are the cyclic extensions of order p? Can you construct extensions with a given Galois group? Give an example of a Galois group (presumably over Q) other than A_n or S_n. Why can you not trisect an angle with straightedge and compass? What numbers are constructible by straightedge-and-compass construction? Which regular n-gons are constructible? Talk about symmetric polynomials... If F:K has only finitely many intermediate extensions, show that it is a simple extension. Prove the fundamental theorem of algebra. How many irreducible polynomials of degree six are there over F_2? Suppose Gal(F/Q) is quaternion. How many quadratic subextensions in F? Can these be imaginary? Division rings -------------- State and prove Wedderburn's theorems (about division rings). Give an example of a division ring which isn't a field. What is the group of unit quaternions topologically? How does it relate to SO(3)? Can a polynomial over a division ring have more roots than its degree? Classify real division algebras. Structure Theorems for Modules/PIDS and f.g. Abelian groups ----------------------------------------------------------- State and prove the structure theorem for abelian groups. If M is free abelian, how can I put the quotients of M in some standard form? Subgroups of a free abelian group? Subgroups of a free group? State the structure theorem for Modules over a PID, and how it applies to linear operators on a vector space. Apply this result to operators p(D), D=d/dx. What is the Jordan Canonical Form of a matrix? What is rational canonical form? What conditions are needed on the field for Jordan form? Give the 4x4 Jordan forms with minimal polynomial (x-1)(x-2)^2. Under what conditions on B can we solve e^A=B, A,B square matrices? If M is free abelian, how can I put quotients of M in some standard form? Representation Theory --------------------- Definition of semisimple algebra. Structure theorem for semisimple algebras (Artin-Wedderburn.) State and prove Maschke's Theorem. How does one prove a f.d. representation of a Lie group is equivalent to a unitary one? Define matrix algebras. State and prove Schur's Lemma. Group theory ------------ Prove A_n is generated by 3-cycles. Prove S_p is generated by a transposition and a p-cycle, if p is prime. Talk about normal subgroups of S_n for n>=5. Simplicity of A_n for n>=4. Find all normal subgroups of A_4. What are the conjugacy classes of S_n? Which groups have nontrivial isomorphisms? Suppose you have a finite p-group and you have a representation of this group on a finite dimensional vector space over a finite field of characteristic p. What can you say about it? Write down a nontrivial example. Show that every such irreducible representation is one dimensional. Name some simple groups. What is the quaternion group? IS there a way for Z_7 to act on a group of order 8? Give an example of an infinite group that is not the product of cycles. Group representations --------------------- Define a group ring. What do you know about them? What does the group ring of Z_5 over Q look like? Now replace Z_5 by the Klein 4-group and the quaternionic group. What is a representation? An irreducible representation? What kind of matrices can be in a representation? What do you know about group homology? IS the character matrix of a finite group always square? Why? Construct the character table of S_5. Sylow subgroups --------------- State Sylow's theorems. Prove them. Groups of order 8,15,35,55. Abelian groups of order 36. Why can you conclude a group of order 35 is the product of its Sylow subgroups? When do two normal subgroups commute? Let G=GF(3,p). What does basic group theory tell us about G? How many conjugates does the p-Sylow have? Give a matrix form for the elements in the Sylow subgroup. Explain the conjugacy (between Sylow subgroups) in terms of eigenvalues and eigenvectors. Give a matrix form for the normalizer of the Sylow p-subgroup. Number theory ------------- What is a ring of integers? What does integral over Z mean? What is the integral closure of Z in Q(i)? Find all prime numbers in Z[i]. Do you know about Dedekind domains and class numbers? Define p-adics. Is Z[t]/(t^p-1)->Z(w), where w is a pth root of unity, an isomorphism? Example of local rings. Completions. Give an example of a PID with a unique prime ideal. State the Chinese remainder Theorem. Gauss' Lemma. Commutative algebra ------------------- Give an example of a ring which is not a UFD (Z[sqrt[-5]]). Demonstrate non unique factorization. When one passes to ideals why does this problem go away? Example of UFD which is not a PID? Not Euclidean? Classify finite fields. Show that the multiplicative group of a finite field is cyclic. What is the radical? What is a noetherian ring? If I is an ideal in a noetherian ring what is the intersection of I^n over all positive integers n? ("satisfied for me to show it's zero for various examples e.g. polynomial rings.") State the Hilbert basis Theorem. State Hilbert's Nullstellensatz. Talk about factorization in a polynomial rings. For what R is it true that R[x_1,...x_m] is a UFD? What's wrong with unique factorization if we don't have a domain? Can you have a UFD that is not noetherian? Lie groups ---------- What is a Lie group? What is a unitary representation of a Lie group? What is its Lie algebra? The Jacobi identity? What is the adjoint representation of a Lie group? What is the Peter-Weyl theorem. Modules ------- What is a module? A projective module? What are projective modules good for? What is the Ext-functor? Give a nontrivial example. Can you compute Ext^i(Z/2,Z/2) over Z/4 for all i>=0? Compute Ext^1(Z,Z/2) and Ext^1(Z/2,Z). What is the nontrivial extension corresponding to the nontrivial element of Ext^1(Z/2,Z). can you have Ext without having projective resolutions (universal delta functors.) What is finite projective dimension? What is finite injective dimension? What rings beside fields have finite injective dimension. Why is (x^p-1)/(x-1) irreducible over Q? What is the Eisenstein criterion? Miscellaneous ------------- Talk about isomorphism classes of subgroups of Q. How many are there?