~~META:title=Previous Homework~~

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===== Math 402 - 01 Previous Homework (Spring 2019)=====
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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 $\aut(E)$, $F=E_G$, 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 $L,L_1,L_2\in\sub_F(E)$  
    - If $H_1 \leq H_2$, then $H_2^* \leq H_1^*$. (i.e. $\,^*$ is order reversing)
    - If $L_1 \leq L_2$, then $L_2^* \leq L_1^*$.  (i.e. $\,^*$ is order reversing)
    - $H\leq H^{**}$ (i.e. $1 \leq \,^{**}$)
    - $L\leq L^{**}$ (i.e. $1 \leq \,^{**}$)
  - 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, $\alpha_1,\dots,\alpha_n$ 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 $G=\aut_\Q(E)$.
    - What is the order of $G$?
    - Is it abelian?  
    - 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 $f(x)=x^p-t\in F[x]$.  
    - Show that $f(x)$ has no root in $F$.
    - Show that the Frobeni\us endomorphism $\Phi:F\to F$ 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 $f(x), g(x)\in F[x]$. 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 $\varphi:F\to F$ 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
    - $B$ 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 $\dim(W) \leq \dim(V)$.
    - 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 $\{\alpha(v)\mid v\in B\}$ is linearly independent in $W$ iff $\widehat{\alpha}$ is injective.
    - 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. 
    - Prove that if $a,b$ are relatively prime and $a|bc$ then $a|c$.
    - 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: 
    - 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 $\delta(r) < \delta(d))$ 
    - 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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