Reference no: EM131107192
Honors Exam 2010 Algebra
1. Let G be a nonabelian group of order 28 whose Sylow 2 subgroups are cyclic.
(a) Determine the numbers of Sylow 2-subgroups and of Sylow 7-subgroups.
(b) Determine the numbers of elements of G of each order.
(c) Prove that there is at most one isomorphism class of such groups.
2. This problem is about the space V of real polynomials in two variables x, y. The fact that it is an infinite-dimensional space plays a role only in part (c).
If f is a polynomial, ∂f will denote the operator f(∂/∂x, ∂/∂y), and ∂f(g) will denote the result of applying this operator to a polynomial g. We also have the operator of multiplication by f, which we write as mf. So mf(g) = fg.
The rule (f, g) = ∂f(g)0 defines a bilinear form on V , the subscript 0 denoting evaluation of a polynomial at the origin. For example, (x2, x3) = ∂2x(x3)0 = (6x)0 = 0.
(a) Prove that this form is symmetric and positive definite, and that the monomials xiyj form an orthogonal basis of V (not an orthonormal basis).
(b) Linear operators A and B on V are adjoint if (Ap, q) = (p, Bq) for all polynomials p and q. Prove that ∂f and mf are adjoint operators.
(c) When f = x2 + y2, the operator ∂f is the Laplacian, which is often written as ?. A polynomial h is harmonic if ?h = 0. Let be the space H of harmonic polynomials. Identify the space H⊥ orthogonal to H with respect to the given form.
3. Do either (a) or (b).
(a) Describe the maximal ideals in Z[x].
(b) Determine the number of irreducible polynomials of degree 4 in F3[x].
4. Do either (a) or (b).
(a) Let ?: Z[x] → C be the homomorphism that sends x to a complex number γ. Prove that the kernel of ? is a principal ideal.
(b) Let f(x) = x5 + cx4 + a3x3 + a2x2 + a1x + a0 be an integer polynomial such that ai ≡ 0 modulo 3 for all i and that a0 0 modulo 9. There is no condition on the coefficient c of x4 other than that it is an integer. Prove that f is irreducible in Z[x] unless it has an integer root.
5. Do any two of the three parts.
Let R denote the ring of Gauss integers: R = {a + bi | a, b ∈ Z}.
(a) A Gauss prime is a Gauss integer that has no proper Gauss integer factor and is not a unit in R. Factor 3 + 9i into Gauss primes.
(b) Let M denote the additive group (Z/5Z)+. In how many ways can M be given the structure of an R-module?
(c) Solve AX = B for X in R2, when
6. Let α = 3√2, β = √3, and γ = α + β. Let L be the field Q(α, β), and let K be the splitting field of the polynomial (x3 - 2)(x2 - 3) over Q.
(a) Determine the degrees [L: Q] and [K: Q].
(b) Determine all automorphisms of the field L.
(c) Determine the degree of γ over Q.
(d) Let f be the irreducible polynomial for γ over Q. What are the complex roots of f?
(e) Determine the Galois group of K/Q.
Fundamental models in physics-ecology
: See the code in Example 9.13 for generating a simple random walk. Write a function for simulating a biased random walk where the probability of moving left and right is p and 1 - p, respectively. Graph your function obtaining pictures like Figure ..
|
What are the symptoms and long term prognoses
: What are the symptoms and long term prognoses for a child born with FAS - what are the risks to the baby if the mother suffers from chronic stress throughout the pregnancy?
|
What types of change control would be appropriate for large
: What types of change control would be appropriate for small IT projects? What types of change control would be appropriate for large ones?
|
Which partys stance on the issue you selected
: Thinking about your own Political Ideology that we discussed in the introduction discussion, which party's stance on the issue you selected is closest to your own? Which is the furthest from your ideals? Why?
|
Determine all automorphisms of the field l
: Let α = 3√2, β = √3, and γ = α + β. Let L be the field Q(α, β), and let K be the splitting field of the polynomial (x3 - 2)(x2 - 3) over Q. Determine the degrees [L: Q] and [K: Q]. Determine all automorphisms of the field L
|
Cement hydration products
: Find and summarise the names and formulae of four of the major hydration products recognising the variable water content and the use of different notation by cement chemists. Which of the products forms the main component of the cement gel? Find a..
|
Following code was used to generate the graphs
: The following code was used to generate the graphs in Figure 9.3. Modify the code to illustrate the strong law of large numbers for an i.i.d. sequence with the following distributions: (i) Pois(λ = 5), (ii) Norm(-4, 4), (iii) Exp(λ = 0.01).
|
Concentration of glucose in his blood
: John has a fasting blood glucose level of 184 mg/dL. What is the concentration of glucose in his blood. (glucose MW = 180g/mol, mg/dL = mg/100 mL)?
|
What are teratogens
: What are Teratogens? They are anything that will interfere with the growth or development of a fetus.
|