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+ | ~~META:title=Previous Homework~~ | ||
+ | <WRAP centeralign> | ||
+ | ===== Math 402 - 01 Previous Homework (Spring 2019)===== | ||
+ | </WRAP> | ||
+ | {{page>people:fer:402ws:402ws_homework_header&nofooter&noeditbtn}} | ||
+ | ---- | ||
+ | {{page>people:fer:402ws:defs&nofooter&noeditbtn}} | ||
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+ | **Problem Set 07** (complete) Due: 04/17/2019 Board presentation: 04/26/2019 | ||
+ | - Let E be a field, G a finite subgroup of \aut(E), F=EG, and L∈\subF(E). Show that L∗=\autL(E), and it is a subgroup of G. | ||
+ | - Let E be a field, G a subgroup of \aut(E), and F=EG. Prove that for any H,H1,H2∈\sub(G), and any L,L1,L2∈\subF(E) | ||
+ | - If H1≤H2, then H∗2≤H∗1. (i.e. ∗ is order reversing) | ||
+ | - If L1≤L2, then L∗2≤L∗1. (i.e. ∗ is order reversing) | ||
+ | - H≤H∗∗ (i.e. 1≤∗∗) | ||
+ | - L≤L∗∗ (i.e. 1≤∗∗) | ||
+ | - Let E/L/F be a field tower. | ||
+ | - Prove that if E/F is a normal extension then so is E/L. | ||
+ | - Prove that if E/F is a Galois extension then so is E/L. | ||
+ | |||
+ | **Problem Set 06** (complete) Due: 04/12/2019 Board presentation: 04/17/2019 | ||
+ | - Let F be a field, α1,…,αn elements from some extension E of F, and R a commutative ring with unity. If φ1,φ2:F(α1,…,αn)→R are homomorphisms such that φ1(a)=φ2(a) for all a∈F and φ1(αi)=φ2(αi) for i=1,…,n, then φ1=φ2. | ||
+ | - Let f(x)=x5−2∈\Q[x], and E the splitting field of f(x). Consider the group G=\aut\Q(E). | ||
+ | - What is the order of G? | ||
+ | - Is it abelian? | ||
+ | - What are the orders of elements in G? | ||
+ | - Let F=\Fp(t) be the field of rational functions on t with coefficients in \Fp. Consider the polynomial f(x)=xp−t∈F[x]. | ||
+ | - Show that f(x) has no root in F. | ||
+ | - Show that the Frobeni\us endomorphism Φ:F→F is not surjective. | ||
+ | - Show that f(x) has exactly one root, and that root has multiplicity p. | ||
+ | - Show that f(x) is irreducible over F. | ||
+ | |||
+ | **Problem Set 05** (complete) Due: 03/25/2019 Board presentation: 04/02/2019 | ||
+ | - Let F be a field and f(x),g(x)∈F[x]. Prove: | ||
+ | - (f(x)+g(x))′=f′(x)+g′(x) | ||
+ | - (f(x)g(x))′=f(x)g′(x)+f′(x)g(x) | ||
+ | - Let F be a field, and φ:F→F an endomorphism of F. Prove that the set Fφ={a∈F∣φ(a)=a} is a subfield of F. | ||
+ | - How many monic irreducible polynomials of degree 4 are there over \F5? | ||
+ | - Let E be a field extension of F. Prove that E is an algebraic closure of F iff E is minimal with the property that every polynomial f(x)∈F[x] splits over E. | ||
+ | |||
+ | **Problem Set 04** (complete) Due: 03/11/2019 Board presentatiion: 03/25/2019 | ||
+ | - Let E/F be a field extension. Prove that [E:F]=1 iff E=F. | ||
+ | - Let E and K be field extensions of F and φ:E→K an F-extension homomorphism. Show that φ is a linear transformation of F-vector spaces. | ||
+ | - Write \sq2 as a polynomial expression on α=\sq2+\sq3. | ||
+ | - Find the minimal polynomial of u=(\sq[3]2+ω) over \Q. | ||
+ | |||
+ | |||
+ | **Problem Set 03** (complete) Due: 02/18/2019 Board presentation: 02/20/2019 | ||
+ | - Let V be a vector space and B⊆V. Show that the following are equivalent | ||
+ | - B is a basis for V, | ||
+ | - B is maximal linearly independent set, | ||
+ | - B is minimal spanning set. | ||
+ | - Let V be a vector space and W a subspace of V. | ||
+ | - Prove that dim(W)≤dim(V). | ||
+ | - Prove that if V is finite dimensional and dim(W)=dim(V) then W=V | ||
+ | - Show, with a counterexample, that the finite dimensional hypothesis is necessary in part b. | ||
+ | - In regards to the //Universal Mapping Property// for vector spaces discussed in class today: | ||
+ | - Complete the proof of it. | ||
+ | - Prove that the set {α(v)∣v∈B} is linearly independent in W iff ˆα is injective. | ||
+ | - Prove that the set {α(v)∣v∈B} is a spanning set for W iff ˆα is surjective. | ||
+ | - Let V be a vector space over F, and W a subspace of V. Let B1 be a basis for W and B a basis for V such that B1⊆B. Prove that the set {v+W∣v∈B−B1} is a basis for the quotient space V/W. | ||
+ | |||
+ | **Problem Set 02** (complete) Due: 02/11/2019 Board presentation: 02/18/2019 | ||
+ | - Let D be a UFD. a,b,c∈D, and f(x)∈D[x]. a,b are said to be "//relatively prime//" if gcd(a,b) is a unit. | ||
+ | - Prove that if a,b are relatively prime and a|bc then a|c. | ||
+ | - Prove that if ab is a root of f(x), and a,b are relatively prime, then a divides the constant term of f(x) and b divides the leading term of f(x). | ||
+ | - Let D be an ED, a,b∈D, with b≠0. Consider the sequence r0,r1,r2,…,rn defined recursively as follows: r0=a,r1=b, and using the propery of an Euclidean Domain, until obtaining a residue 0, \[ | ||
+ | \begin{array}{rclll} | ||
+ | r_0 &=&q_1 r_1 + r_2 &\text{ and} &\delta(r_2) < \delta(r_1), \\ | ||
+ | r_1 &=&q_2 r_2 + r_3 &\text{ and} &\delta(r_3) < \delta(r_2), \\ | ||
+ | &\vdots \\ | ||
+ | r_{n-3} &=&q_{n-2} r_{n-2} + r_{n-1} &\text{ and} &\delta(r_{n-1}) < \delta(r_{n-2}), \\ | ||
+ | r_{n-2} &=&q_{n-1} r_{n-1} + r_n &\text{ and} &r_n=0. \\ | ||
+ | \end{array} | ||
+ | \] Why does the sequence r1,r2,…,rn have to eventually attain the value rn=0? Prove that the last non-zero entry in the residues list, i.e. rn−1∼gcd(a,b). | ||
+ | - Let D be a PID, a,b∈D. Let d be a generator of the ideal \pbra+\pbrb. Show that d∼gcd(a,b). | ||
+ | - Let D be an ID, a,b∈D. Prove that if a and b have a least common multiple l∈D, then abl is a greatest common divisor of a and b in D. | ||
+ | - (Optional) Let γ=\ds1+√−192 and consider the subring of \C given by: R={a+bγ∣a,b∈\Z} Prove that R is a PID but not an ED. A detailed proof can be found in Mathematics Magazine, Vol. 46, No. 1 (1973), pp 34-38. If you choose to work on this problem, do not consult this reference, or any other reference. Hand-in only your own work, even it it is only parts of the solution. | ||
+ | |||
+ | |||
+ | **Problem Set 01** (complete) Due: 02/01/2019 Board presentation: 02/08/2019 | ||
+ | - Let D be an integral domain. Consider the following two properties that D and a function δ:D−{0}→\N0 may have: | ||
+ | - For any a,d∈D with d≠0, there are q,r∈D such that \\ a=qd+r and ( r=0 or δ(r)<δ(d)) | ||
+ | - For any a,b∈D−{0}, δ(a)≤δ(ab). \\ Prove that if there is a function δ satisfying the first condition, then there is a function γ satisfying both of them. Hint: consider γ defined by: γ(a):=minx∈D−{0}δ(ax) | ||
+ | - Chapter 18, problem 22. | ||
+ | - Chapter 16, problem 24. Can you weaken the assumption "infinitely many"? | ||
+ | - Show that an integral domain D satisfies the ascending chain condition ACC iff every ideal of D is finitely generated. (Hint: one direction is similar to the proof that every PID satisfies the ACC). | ||
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