Department of Mathematics and Statistics, Binghamton University people:fer:402ws:spring2019

Modern Algebra II - Math 402 - 01 (Spring 2019)

2019-05-09T09:37:21-04:00

Modern Algebra II - Math 402 - 01 (Spring 2019)

Spring 2019

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Instructor:	Fernando Guzmán	WH-116	x-72876	fer@math.binghamton.edu

Classroom:	WH-329	MWF	9:40 - 10:40
Classroom:	WH-329	T	8:30 - 9:55

Office Hours:	Monday	1:15 - 2:15	p.m.
(subject to change)	Tuesday	4:00 - 5:00	p.m.
(subject to change)	Friday	8:30 - 9:30	a.m.

Announcements

Class notes for the periods 01/22/19-02/06/19 and 02/11/19-02/19/19 are posted below. They are likely to contain typos and mistakes. Please send me a note whenever you see one.

Homework

Class notes 01/22/19-02/06/19

Class notes 02/11/19-02/19/19

Class notes 02/19/19-03/15/19

Class notes 03/25/19-04/12/19

Class notes 04/15/19-05/10/19

Homework

2019-05-09T09:11:58-04:00

Math 402 - 01 Homework (Spring 2019)

Problem Set 10 (complete) Due: 05/10/2019

Let $F$ be a field of characteristic zero, $a\in F$, and $\xi=\xi_n$ a primitive $n$-th root of unity.
1. Show by example that $\gal_F(F(\xi))$ need not be all of $U_n$.
2. Show by example that $\gal_{F(\xi)}(F(\xi,\sq[n]{a}))$ need not be all of $C_n$.
Let $G$ and $H$ be solvable groups. Prove that $G\times H$ is solvable.
Show that the change of variable $y=x+(a/3)$ transforms the general cubic equation \[ x^3+ax^2+bx+c = 0 \] into a depressed cubic. Therefore, Cardano's formula is useful to solve any cubic equation.

Problem Set 09 (complete) Due: 05/03/2019 Board presentation: 05/10/2019

Prove that the homomorphism \[ \begin{array}{rccc} \psi:& U_n &\to &\gal(\Q(\xi_n)/\Q) \\ & k &\mapsto &\psi_k \\ \end{array}\] is surjective and injective.
Let $\xi_{15}=\cis(2\pi/15)$ be a primitive $15$-th root of unity.
1. Find the group $\gal(\Q(\xi_{15})/\Q)$ and draw its lattice of subgroups.
2. Find and draw the lattice of intermediate fields of the extension $\Q(\xi_{15})/\Q$.
3. Write down the correspondence between the subgroups in part 1, and the subfields in part 2, using the Fundamental Theorem of Galois Theory.
Show that any non-abelian simple group is non-solvable.
Show that if $d$ is a divisor of $n$ then $\Q(\xi_d)$ is a subfield of $\Q(\xi_n)$. Conclude that $\varphi(d)$ divides $\varphi(n)$, and $U_d$ is a quotient of $U_n$.

Problem Set 08 (complete) Due: 04/26/2019 Board presentation: 05/03/2019

Prove the following corollary to the Fundamental Theorem of Galois Theory. Use only the FTGT statements to prove it. Let $E/F$ be a (finite) Galois extension, with Galois group $G=\gal_F(E)$. Let $L_1,L_2\in\sub_F(E)$ and $H_1,H_2\in\sub(G)$.
1. $(L_1\meet L_2)^* = L_1^* \join L_2^*$
2. $(L_1\join L_2)^* = L_1^* \meet L_2^*$
3. $(H_1\meet H_2)^* = H_1^* \join H_2^*$
4. $(H_1\join H_2)^* = H_1^* \meet H_2^*$
Let $f(x)\in\Q[x]$ be such that it has a non-real root. Let $E$ be the splitting field of $f(x)$ over $\Q$. Prove that $\gal_\Q(E)$ has even order.
Consider the polynomial $f(x)=x^3+2x^2+2x+2\in\Q[x]$, and $E$ its splitting field over $\Q$.
1. Show that $f(x)$ has exactly one real root. (Hint: use calculus)
2. Show that $f(x)$ is irreducible over $\Q$.
3. Find $[E:\Q]$. Fully explain your calculation.
4. Determine $\gal_\Q(E)$.
Consider the group $S_n$ of all permutations of the set $\{1,2,\dots,n\}$.
1. Show that the transpositions $(1\ \ 2),(2\ \ 3),\dots,(n-1\ \ n)$ generate the whole group $S_n$.
2. Show that $S_n$ is generated by the following two permutations: \[ \rho = (1\ \ 2\ \ \dots\ \ n) \quad \text{and} \quad \sigma=(1\ \ 2) \] (Hint: conjugate $\sigma$ by $\rho$.)
3. For $p$ is a prime, $\rho$ a $p$-cycle, and $\sigma$ a transposition, show that $\rho$ and $\sigma$ generate $S_p$. Show, by counterexample, that the hypothesis of $p$ being prime cannot be removed.

Previous Homework

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Previous Homework

2019-04-30T14:18:16-04:00

Math 402 - 01 Previous Homework (Spring 2019)

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

Let $E$ be a field, $G$ a finite subgroup of $\aut(E)$, $F=E_G$, and $L\in\sub_F(E)$. Show that $L^*=\aut_L(E)$, and it is a subgroup of $G$.
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. $1 \leq \,^{**}$)
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

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 $\varphi_1=\varphi_2$.
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 $G$?
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

Let $F$ be a field and $f(x), g(x)\in F[x]$. Prove:
1. $(f(x)+g(x))' = f'(x) + g'(x)$
2. $(f(x)g(x))' = f(x)g'(x) + f'(x)g(x)$
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 $F$.
How many monic irreducible polynomials of degree 4 are there over $\F_5$?
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

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 $\varphi:E\to K$ an $F$-extension homomorphism. Show that $\varphi$ is a linear transformation of $F$-vector spaces.
Write $\sq{2}$ as a polynomial expression on $\alpha=\sq{2}+\sq{3}$.
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

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.
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.
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.
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

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 $f(x)$.
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. $r_{n-1}\sim\gcd(a,b)$.
Let $D$ be a PID, $a,b\in D$. Let $d$ be a generator of the ideal $\pbr{a}+\pbr{b}$. Show that $d\sim\gcd(a,b)$.
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 $D$.
(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

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: \[ \gamma(a):= \min_{x\in D-\{0\}}\delta(ax)\]
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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