Homework for Introduction to Algebra II, M.Math

Practise problems for group theory

Week 1

• From Dummit and Foote, section 1.6: 4, 25.
• From Dummit and Foote, section 2.2: 7.
• From Dummit and Foote, section 3.1: 36.
• From Dummit and Foote, section 3.2: 18.
• From Dummit and Foote, section 3.3: 7.

  Week 2

• From Dummit and Foote, section 3.5: 3, 12.
• From Dummit and Foote, section 4.1: 8.
• From Dummit and Foote, section 4.2: 4, 8, 14.
• From Dummit and Foote, section 4.3: 5, 35.
• From Dummit and Foote, section 4.5: 16, 23.

  Week 3

• From Dummit and Foote, section 4.4: 18.
• From Dummit and Foote, section 5.4: 19.
• From Dummit and Foote, section 5.5: 6, 8, 24.

  Week 4

• From Dummit and Foote, section 6.3: 3, 10.
• From Dummit and Foote, section 13.1: 5, 8.
• From Dummit and Foote, section 13.2: 4, 5, 17.

  Week 5

• From Dummit and Foote, section 13.2: 22.
• From Dummit and Foote, section 13.4: 3, 4. 5.

  Week 6

• From Dummit and Foote, section 09.4: 16
• From Dummit and Foote, section 13.5: 2, 5, 11.
• From Dummit and Foote, section 14.1: 4, 6, 7 (though Aut(C/Q) is an infinite group).

  Week 7

• From Dummit and Foote, section 13.6: 3, 9
• From Dummit and Foote, section 14.2: 6, 12, 31.
• From Dummit and Foote, section 14.3: 8.
• From Dummit and Foote, section 14.4: 1.
• From Dummit and Foote, section 14.5: 5, 10.

  Week 8

• From Dummit and Foote, section 14.6: 3, 7, 12.
• From Dummit and Foote, section 14.7: 7, 12, 17.
• Let $p$ be a prime number. Show that a subgroup of $S_p$ containing a $p$-cycle and a transposition is $S_p$. Write a polynomial $f(x)$ of degree $p$ with integer coefficients such that the Galois group of the splitting field of $f(x)$ over $\mathbb{Q}$ is $S_p$.

Week 9

• From Dummit and Foote, section 13.3: 2, 4.
• Let a real rumber $b$ be constructible by straight edge and compass. Show that the Galois group of the Galois closure of $\mathbb{Q}(b)/\mathbb{Q}$ is a solvable group.
• Show that $K/F$ is purely inseparable iff there is a unique $F$-embedding of $K$ into the algebraic closure of $F$. Show that purely inseparable extensions are normal extension.
• Let $F\subset K\subset L$ be fields such that $L/F$ is a finite extension. Let $x\in L$, show the following trace and norm formulas:
  1) $T_{K/F}(T_{L/K}(x))=T_{L/F}(x)$
  2) $N_{K/F}(N_{L/K}(x))=N_{L/F}(x)$