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people:fer:402ws:spring2019:previous_homework [2019/04/12 08:09]
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people:fer:402ws:spring2019:previous_homework [2019/04/30 14:18] (current)
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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}}
 + 
 +**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 H1H2, then H2H1. (i.e. is order reversing)
 +    - If L1L2, then L2L1. ​ (i.e. is order reversing)
 +    - HH (i.e. 1)
 +    - LL (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 aF and φ1(αi)=φ2(αi) for i=1,,n,​ then φ1=φ2. ​
 +  - Let  f(x)=x52\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)=xptF[x].  ​
 +    - Show that f(x) has no root in F.
 +    - Show that the Frobeni\us endomorphism Φ:FF 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:FF an endomorphism of F. Prove that the set Fφ={aFφ(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:EK 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 BV.  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)vB} is linearly independent in W iff ˆα is injective.
 +    - Prove that the set {α(v)vB} 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 B1B.  Prove that the set {v+WvBB1} 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,cD, 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,bD, with b0. 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. rn1gcd(a,b).
 +  - Let D be a PID, a,bD. Let d be a generator of the ideal \pbra+\pbrb. Show that dgcd(a,b).
 +  - Let D be an ID, a,bD.  Prove that if a and b have a least common multiple lD, 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+bγa,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,dD with d0, there are q,rD such that \\ a=qd+r ​ and ( r=0 or δ(r)<δ(d)) ​
 +    - For any a,bD{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):=minxD{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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