Homework for Introduction to Algebra II, M.Math

Week 1

• From Dummit and Foote, section 09.4: 1(c), 11, 16, 20(a).
• From Dummit and Foote, section 13.1: 1, 5, 8.

  Week 2

• From Dummit and Foote, section 13.2: 1, 4, 5, 12, 17.
• From Dummit and Foote, section 13.4: 3, 4.

  Week 3

• From Dummit and Foote, section 13.4: 5.
• From Dummit and Foote, section 13.5: 2, 5, 8, 11.

  Week 4

• From Dummit and Foote, section 13.6: 3, 9
• From Dummit and Foote, section 14.1: 4, 6, 7(though Aut(C/Q) is an infinite group), 10.
• Show that the cyclotomic polynomials are irreducible.

  Week 5

• From Dummit and Foote, section 14.2: 1, 3, 6, 12, 31.
• From Dummit and Foote, section 14.3: 3, 8.
• From Dummit and Foote, section 14.4: 1.
• From Dummit and Foote, section 14.5: 5, 10.

  Week 6

• From Dummit and Foote, section 14.6: 3, 7, 12.
• From Dummit and Foote, section 14.7: 7, 12, 17.

  Week 7

• Show that $\widehat{\mathbb{Z}}=\Pi_{p \text{ primes}}\widehat{\mathbb{Z}_p}$. In particular $\widehat{\mathbb{Z}}$ is uncountable.
• Let $(G_i, i\in I)$ be an inverse system of finite groups with discerete topology. Show that the inverse limit $G$ has a basis consisting of $\phi_i^{-1}(g_i)$ where $\phi_i:G\to G_i$ is the natural map and $g_i\in G_i$ for $i\in I$.
• Let $K/F$ be Galois extension (possibly infinite) and $L\subset K$ be a finite Galois extension of $F$. Show that $Gal(K/L)$ is a normal subgroup of $Gal(K/F)$ and the quotient is isomrphic to $Gal(L/F)$.
• In the above, let $H\le Gal(K/F)$, $M=K^H$ and $\tilde H=Gal(K/M)$. Let $U$ be an open normal subgroup of $Gal(K/F)$ containing $H$. Then show that $\tilde H \subset U$. Use this to show that $\tilde H$ is the closure of $H$ in $Gal(K/F)$.

Week 8

• Let $G=\Gal(\mathbb{Q}(\sqrt{p}; p \text{ prime })/\mathbb{Q})$. Show that $G$ has a finite index subgroup which is not open. (Hint: Let $H$ be the subgroup consisting of automorphism which is identity on all but finitely many $\sqrt{p}$ as $p$ varies over all primes. Show that $G$ has structure of a $\FF_2$ vector space and use it to show there exist a finite index subgroup of $G$ containing $H$.)
• Show that purely inseparable extensions are normal extension.
• Show that $K/F$ is purely inseparable iff there is a unique $F$-embedding of $K$ into the algebraic closure of $F$.