University of Wisconsin-Madison Mathematics Department

Math 541 Modern Algebra A first course in Abstract Algebra Lecturer: Arun Ram

Fall 2007

Homework 2: Due September 19, 2007

1. Define monoid without identity, monoid, group, ring without identity, ring, division ring and field, and give examples. Make sure that your example of a monoid without identity is not a monoid, that your example of a monoid is not a group, etc.

2. Give and example of an operation on $\mathbb{Z}$ that is not associative.

3. Let $G$ be a group. Show that the identity element of $G$ is unique.

4. Let $G$ be a group and let $g\in G$. Show that the inverse of $g$ is unique.

5. Why isn't $\{0,1,2,3,4,5\}$ a group?

6. Show that $–(–5)=5$.

7. Show that $1/(1/5)=5$.

8. Show that $–1\cdot 5=–5$.

9. Show that $0\cdot 5=0$.

10. Define $\mathbb{Q}$ and prove that it is a field.

11. Define the quaternions and show that they are a division ring and not a field.

12. Define $\mathbb{Z}/n\mathbb{Z}$ and prove that it is a group.

13. Define $\mathbb{Z}/n\mathbb{Z}$ and prove that it is a ring.

14. For which positive integers $n$ is $\mathbb{Z}/n\mathbb{Z}$ a field?

15. Let $n$ be a positive integer. An $n$th root of unity is a complex number $a$ such that $a^n=1$.

    Let $C_n$ be the set of $n$th roots of unity in $ℂ$. Determine $C_3$, $C_4$, $C_5$, and graph these sets.

16. Let $C_n$ be the set of $n$th roots of unity in $ℂ$. Show that $C_n$ is a group.

17. Define $M_n\left(ℂ\right)$ and prove that it is a ring.

18. Define $ℂ\left[x\right]$ and prove that it is a ring.

19. Define $ℂ\left(x\right)$ and prove that it is a field.

20. Define $ℂ\left[\right[x\left]\right]$ and prove that it is a ring.

21. Define $ℂ\left(\right(x\left)\right)$ and prove that it is a field.

22. Show that each element of $ℂ\left(\right(x\left)\right)$ has a unique expression in the form $x^{\ell }p$, where $p\in ℂ\left[x\right]$ and has nonzero constant term.

23. Show that there exists a field with 4 elements.