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## Math 402 - 01 Homework (Spring 2019)


Problem Set 08 (partial) Due: 04/26/2019 Board presentation: 04/??/2019

1. 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. $(H_1\join H_2)^* = H_1^* \meet H_2^*$
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.
3. 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 $\gal_\Q(E)$.

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

1. Let $E$ be a field, $G$ a 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 $F$.