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people:fer:401ws:fall2018:homework

## Math 401 - 01 Homework (Fall 2018)

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Problem Set 13 (complete) Due: 12/10/2018, optional (bring to review session)

1. 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 $D(x)=F(x)$.
2. Prove that the operation that defines the external semi-direct product is in fact associative.
3. 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

1. Chapter 14, problems 12, 14. Warning: pay attention to the definition of $AB$.
2. Chapter 14, problem 28. What about the converse?
3. 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

1. Chapter 12, problem 18. Moreover, if $R$ is commutative, then $S$ is an ideal of $R$.
2. Chapter 12, problem 28.
3. Chapter 13, problem 52.
4. Chapter 13, problem 34.