Reference no: EM131083942
Math 176: Algebraic Geometry, Fall 2014- Assignment 2
1 For each of the following, determine if the statement is true or false. If true, give a proof. If false, give a counterexample.
(a) C[x, y] is a PID.
(b) If R is a UFD, and a, b ∈ R such that (a) = (b), then a = ub for some unit u ∈ R.
(c) If R is a PID and I is a prime ideal, then R/I is a PID.
2. (a) Let I = (x2 + 1) in the ring C[x]. Prove C[x]/I ≅ C × C.
(b) Let I = (x2 + 1, x - 2) in Z[x]. Prove Z[x]/I ≅ Z/5Z.
3. Let X = {(t, t2, t3) ∈ A3(C): t ∈ C}. Show that I(X) = J where J = (z - x3, y - x2).
4. For each of the following, determine if the statement is true or false. If true, give a proof. If false, give a counterexample. (You may assume the field k has characteristic zero).
(a) If X, Y ⊆ An, then X = Y if and only if I(X) = I(Y).
(b) If X,Y ⊆ An are algebraic sets, then X = Y if and only if I(X) = I(Y).
(c) If J is an ideal in k[x1, . . . , xn], then rad(J) ⊆ I(V(J)).
(d) If J is an ideal in k[x1, . . . , xn], then rad(J) = I(V(J)).
|
What is the minimum distance that connects all the nodes
: Given the following distances between destination nodes, what is the minimum distance that connects all the nodes?
|
|
Use matlab to simulate a cdma system
: In this problem, we evaluate the bit error rate (BER) performance of the CDMA communications system introduced in Problem 8.3.9. In our experiments, we will make the following additional assumptions.
|
|
What proportion of us men
: (a) What proportion of US men are between 5 ft 7.0 in and 6 ft 1.0 in tall (67.0 and 73.0 inches, respectively)? (b) If a man is at the 14th percentile in height, how tall is he?
|
|
How fast is the ball moving
: A Doppler speed meter operating at exactly 1.02 x105 Hz emits sound waves and detects the same waves after they are reflected from a baseball thrown by the pitcher. How fast is the ball moving?
|
|
Determine if the statement is true or false
: Math 176: Algebraic Geometry, Fall 2014- Assignment 2. For each of the following, determine if the statement is true or false. If true, give a proof. If false, give a counterexample. C[x, y] is a PID
|
|
Approximate score for a student
: (a) What is the estimated percentile for a student who scores 705 on Critical Reading? (b) What is the approximate score for a student who is at the 34th percentile for Critical Reading?
|
|
Determine the folding frequency in the spectrum
: The folding frequency in the spectrum.
|
|
Relative frequency of symbol errors in the simulated
: Simulate the M-ary PSK system for M = 8 and M = 16. Let ERR denote the relative frequency of symbol errors in the simulated transmission in 105 symbols. For each value of M, graph ERR, as a function of the signal-to-noise power ratio (SNR) γ = E/σ2.
|
|
Determining the sections of introductory statistics
: Exam grades across all sections of introductory statistics at a large university are approximately normally distributed with a mean of and a standard deviation of . Use the normal distribution to answer the following questions.
|