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people:fer:402ws:spring2019:previous_homework

Math 402 - 01 Previous Homework (Spring 2019)


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Problem Set 07 (complete) Due: 04/17/2019 Board presentation: 04/26/2019

  1. Let E be a field, G a finite subgroup of \aut(E), F=E_G, and L\in\sub_F(E). Show that L^*=\aut_L(E), and it is a subgroup of G.
  2. Let E be a field, G a subgroup of \aut(E), and F=E_G. Prove that for any H,H_1,H_2\in\sub(G), and any L,L_1,L_2\in\sub_F(E)
    1. If H_1 \leq H_2, then H_2^* \leq H_1^*. (i.e. \,^* is order reversing)
    2. If L_1 \leq L_2, then L_2^* \leq L_1^*. (i.e. \,^* is order reversing)
    3. H\leq H^{**} (i.e. 1 \leq \,^{**})
    4. L\leq L^{**} (i.e. 1 \leq \,^{**})
  3. Let E/L/F be a field tower.
    1. Prove that if E/F is a normal extension then so is E/L.
    2. 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

  1. Let F be a field, \alpha_1,\dots,\alpha_n elements from some extension E of F, and R a commutative ring with unity. If \varphi_1,\varphi_2:F(\alpha_1,\dots,\alpha_n)\to R are homomorphisms such that \varphi_1(a)=\varphi_2(a) for all a\in F and \varphi_1(\alpha_i)=\varphi_2(\alpha_i) for i=1,\dots,n, then \varphi_1=\varphi_2.
  2. Let f(x)=x^5-2\in\Q[x], and E the splitting field of f(x). Consider the group G=\aut_\Q(E).
    1. What is the order of G?
    2. Is it abelian?
    3. What are the orders of elements in G?
  3. Let F=\F_p(t) be the field of rational functions on t with coefficients in \F_p. Consider the polynomial f(x)=x^p-t\in F[x].
    1. Show that f(x) has no root in F.
    2. Show that the Frobeni\us endomorphism \Phi:F\to F is not surjective.
    3. Show that f(x) has exactly one root, and that root has multiplicity p.
    4. Show that f(x) is irreducible over F.

Problem Set 05 (complete) Due: 03/25/2019 Board presentation: 04/02/2019

  1. Let F be a field and f(x), g(x)\in F[x]. Prove:
    1. (f(x)+g(x))' = f'(x) + g'(x)
    2. (f(x)g(x))' = f(x)g'(x) + f'(x)g(x)
  2. Let F be a field, and \varphi:F\to F an endomorphism of F. Prove that the set F_\varphi=\{a\in F\mid\varphi(a)=a\} is a subfield of F.
  3. How many monic irreducible polynomials of degree 4 are there over \F_5?
  4. 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)\in F[x] splits over E.

Problem Set 04 (complete) Due: 03/11/2019 Board presentatiion: 03/25/2019

  1. Let E/F be a field extension. Prove that [E:F]=1 iff E=F.
  2. Let E and K be field extensions of F and \varphi:E\to K an F-extension homomorphism. Show that \varphi is a linear transformation of F-vector spaces.
  3. Write \sq{2} as a polynomial expression on \alpha=\sq{2}+\sq{3}.
  4. Find the minimal polynomial of u=(\sq[3]{2}+\omega) over \Q.

Problem Set 03 (complete) Due: 02/18/2019 Board presentation: 02/20/2019

  1. Let V be a vector space and B\subseteq V. Show that the following are equivalent
    1. B is a basis for V,
    2. B is maximal linearly independent set,
    3. B is minimal spanning set.
  2. Let V be a vector space and W a subspace of V.
    1. Prove that \dim(W) \leq \dim(V).
    2. Prove that if V is finite dimensional and \dim(W)=\dim(V) then W=V
    3. Show, with a counterexample, that the finite dimensional hypothesis is necessary in part b.
  3. In regards to the Universal Mapping Property for vector spaces discussed in class today:
    1. Complete the proof of it.
    2. Prove that the set \{\alpha(v)\mid v\in B\} is linearly independent in W iff \widehat{\alpha} is injective.
    3. Prove that the set \{\alpha(v)\mid v\in B\} is a spanning set for W iff \widehat{\alpha} is surjective.
  4. Let V be a vector space over F, and W a subspace of V. Let B_1 be a basis for W and B a basis for V such that B_1\subseteq B. Prove that the set \{v+W\mid v\in B-B_1\} is a basis for the quotient space V/W.

Problem Set 02 (complete) Due: 02/11/2019 Board presentation: 02/18/2019

  1. Let D be a UFD. a,b,c\in D, and f(x)\in D[x]. a,b are said to be ”relatively prime” if \gcd(a,b) is a unit.
    1. Prove that if a,b are relatively prime and a|bc then a|c.
    2. Prove that if \frac{a}{b} 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).
  2. Let D be an ED, a,b\in D, with b\neq 0. Consider the sequence r_0,r_1,r_2,\dots,r_n defined recursively as follows: r_0=a,r_1=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 r_1,r_2,\dots,r_n have to eventually attain the value r_n=0? Prove that the last non-zero entry in the residues list, i.e. r_{n-1}\sim\gcd(a,b).
  3. Let D be a PID, a,b\in D. Let d be a generator of the ideal \pbr{a}+\pbr{b}. Show that d\sim\gcd(a,b).
  4. Let D be an ID, a,b\in D. Prove that if a and b have a least common multiple l\in D, then \frac{ab}{l} is a greatest common divisor of a and b in D.
  5. (Optional) Let \gamma=\ds\frac{1+\sqrt{-19}}{2} and consider the subring of \C given by: R = \{a + b\gamma\mid a,b\in\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

  1. Let D be an integral domain. Consider the following two properties that D and a function \delta:D-\{0\}\to\N_0 may have:
    1. For any a,d\in D with d\neq 0, there are q,r\in D such that
      a=qd+r and ( r=0 or \delta(r) < \delta(d))
    2. For any a,b\in D-\{0\}, \delta(a)\leq\delta(ab).
      Prove that if there is a function \delta satisfying the first condition, then there is a function \gamma satisfying both of them. Hint: consider \gamma defined by: \gamma(a):= \min_{x\in D-\{0\}}\delta(ax)
  2. Chapter 18, problem 22.
  3. Chapter 16, problem 24. Can you weaken the assumption “infinitely many”?
  4. 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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people/fer/402ws/spring2019/previous_homework.txt · Last modified: 2019/04/30 14:18 by fer