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===== Math 401 - 01 Homework (Fall 2018)=====
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**Problem Set 13** (complete) Due: 12/10/2018, optional (bring to review session)
  - Let $D$ be an I.D., $D[x]$ the ring of polynomials over $D$, and $D(x)$ the field of rational functions over $D$.  Let $F$ be the field of fractions of $D$, $F[x]$ the ring of polynomials over $F$, and $F(x)$ the field of rational functions over $F$.  Show that $D(x)=F(x)$.
  - Prove that the operation that defines the external semi-direct product is in fact associative. 
  - Prove that the two non-abelian semi-direct products of $C_7$ with $C_3$ are isomorphic. (Hint: use the homomorphism discussed in class, given by: $a\mapsto u, b\mapsto v^{-1}$)

**Problem Set 12** (complete) Due: 12/03/2018.  Board presentation: 12/07/2018
  - Chapter 14, problems 12, 14.  Warning: pay attention to the definition of $AB$.
  - Chapter 14, problem 28.  What about the converse?
  - Write $n\in\Z$ as $md_0$, where $d_0$ is the last digit (base 10) and $m$ consists of all other digits. In other words, $n=10 m+d_0$.  Prove that $n$ is divisible by $7$ iff $m-2d_0$ is divisible by $7$. 

**Problem Set 11** (complete) Due: 11/20/2018.  Board presentation: 11/27/2018
  - Chapter 12, problem 18. Moreover, if $R$ is commutative, then $S$ is an ideal of $R$.
  - Chapter 12, problem 28.
  - Chapter 13, problem 52.
  - Chapter 13, problem 34.

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