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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