# 2.4.9

![image-20221017004710321](/home/zack/.config/Typora/typora-user-images/image-20221017004710321.png)



# 3.1.1 de

![image-20221017004756604](/home/zack/.config/Typora/typora-user-images/image-20221017004756604.png)



Rigorous definition of gcd:

![image-20221017005242043](/home/zack/.config/Typora/typora-user-images/image-20221017005242043.png)



Example of division algorithm:

![image-20221017005049934](/home/zack/.config/Typora/typora-user-images/image-20221017005049934.png)

![image-20221017005102458](/home/zack/.config/Typora/typora-user-images/image-20221017005102458.png)

Example of Euclidean algorithm for numbers:

![image-20221017005628608](/home/zack/.config/Typora/typora-user-images/image-20221017005628608.png)

# 3.1.2 bd



![image-20221017004835839](/home/zack/.config/Typora/typora-user-images/image-20221017004835839.png)

# 3.1.15

![image-20221017004917272](/home/zack/.config/Typora/typora-user-images/image-20221017004917272.png)



# Challenge: 3.1.23



![image-20221017005700014](/home/zack/.config/Typora/typora-user-images/image-20221017005700014.png)

![image-20221017010341346](/home/zack/.config/Typora/typora-user-images/image-20221017010341346.png)



- Part b: $x=x^{-1} \implies x^2 = 1 \implies x^2-1 = 0 \implies (x+1)(x-1) = 0\implies x=\pm 1$ since $F$ is a field and hence a domain. Also note $x^2-1$ has at most 2 roots since $F$ is a domain, and at least 2 when $1\neq -1$.
- Part c: write $F^* = \{\alpha, \alpha^2,\cdots, \alpha^{q-1}\}$ for a generator with $q=p^n$.
  Take out all self-inverse elements, i.e. $\pm 1$, these multiply to $-1$.
  The remaining terms all pair up with their inverses and their product is 1, so $\prod \alpha^k = -1$.