Homework 7
MATH 301/601
Complete before Exam 2.
 

Instructions. There will be no test on this assignment; however, the content from this assignment may appear on Exam 2.

Exercise 1.

Let $T$ and $U$ be as in Exercise 9 on HW6. Define $\phi :T\to {ℝ}^{\times }\times {ℝ}^{\times }$ by

1. Show that $\phi$ is a homomorphism.

2. Use the first isomorphism theorem and part (a) to deduce that $T/U$ is isomorphic to ${ℝ}^{\times }\times {ℝ}^{\times }$.

Exercise 2.

Define $\phi :ℝ\to \mathrm{GL}(2,ℝ)$ by

1. Prove that $\phi$ is a homomorphism. (You will need to use the angle-sum formula.)

2. Determine the kernel of $\phi$.

3. The image of $\phi$ is called the 2-dimensional special orthogonal group, denoted $\mathrm{SO}(2)$, and consists of the rotation matrices. Using the first isomorphism theorem, prove that $\mathrm{SO}(2)$ is isomorphic to $ℝ/\mathbb{Z}$.

Exercise 3.

Prove that $ℝ$ and $ℂ$ are not isomorphic as fields. (Surprisingly, they are isomorphic as additive groups!)

Exercise 4.

Let $\phi :F_1\to F_2$ be a homomorphism between fields. Prove that $\phi$ is injective.

Exercise 5.

Let $F$ be a field of characteristic $p\ne 0$. Define $\phi :F\to F$ by $\phi (x)=x^p$.

1. Prove that $\phi$ is a homomorphism.

2. Prove that if $F$ is finite, then $\phi$ is an isomorphism.

Exercise 6.

The goal here is to explore the field of order 9.

1. Find an irreducible quadratic polynomial $p$ in ${\mathbb{Z}}_3[x]$.

2. Let $p$ be your polynomial from part (a). Let $\beta$ be a formal symbol that we declare satisfies $p(\beta )=0$. Then ${𝔽}_9=\{a+b\beta :a,b\in {\mathbb{Z}}_3\text{ and }p(\beta )=0\}$ is a field of order 9. Find the inverses of $1+\beta$, $2+\beta$, and $1+2\beta )$.