Some Math Problems

Try these. If you can't do them, don't worry. That just means we need to slow down the seminar and do more background material. This is fine; we are in now hurry!

1. (Jeff) Does the equation $x^2+2y^2 = -17$ have any solutions with $x,y\in\mathbf{Z}$?

2. (Jennifer) Let $p\in\mathbf{Z}$ be a prime. Prove that $\sqrt{p}$ is irrational.

3. (Mauro) Does the equation $x^2+y^2+z^2 \equiv 7\pmod{8}$ have any solutions with $x,y,z\in\mathbf{Z}$?

4. (Alex) Fermat's Last Theorem asserts that when $n\geq 3$ then $x^n + y^n = z^n$ has no solutions with $xyz\neq 0$. Is the analogue of this statement true when $n=2$?

5. (Jenna) Let $G=\{0,1,2,3\}$ be the group of integers under addition modulo $4$.
   1. What is $2+3$ in $G$?
   2. What is the order of $3$ in $G$?
   3. Let $H=\{1,2,3,4\}$ be the group of nonzero integers under multiplication modulo $5$. Is $G$ isomorphic to $H$? If not, why not? If so, give an explicit isomorphism.

6. (Jeff) What is the tangent line to the graph of $y^2 = x^3 + 3$ at the point $(1,2)$? (Hint: Implicit differentiation.)

7. (Jennifer)
   1. List the elements of a finite field of order $2$.
   2. One can prove that there is a finite field $k$ of order $4$. Does the cubic equation $x^3+x+1=0$ have a solution in $k$?

8. (Mauro)
   1. Prove that the set of elements of finite order in an abelian group is a subgroup.
   2. Prove that a group in which every element except the identity has order $2$ is abelian.

9. (Alexander) Show by example that the product of elements of finite order in a nonabelian group need not have finite order. (Hint: Consider a construction involving $2\times 2$ matrices.)

10. (Jenna) Describe all groups $G$ which contain no proper subgroup.