Math 361, Spring 2021, Assignment 15
From cartan.math.umb.edu
Carefully state the following theorems (you do not need to prove them):[edit]
- Classification of simple field extensions.
Solve the following problems:[edit]
- Find all real solutions of the equation $x^2-5x+3=0$. Be sure to find the exact solutions. Call these solutions $r_1$ and $r_2$.
- Compute the sum $r_1+r_2$, the product $r_1r_2$, and the squares $r_1^2$ and $r_2^2$. Do not make any approximations.
- Find a monomorphism $\phi$ from the quotient ring $\mathbb{Q}[x]/\left\langle x^2-13\right\rangle$ into the real number system.
- Show that the solutions $r_1,r_2$ that you found above both lie in the image of the monomorphism $\phi$ that you defined above, and compute the pre-images $\rho_1=\phi^{-1}(r_1)$ and $\rho_2=\phi^{-1}(r_2)$.
- Working in the quotient ring $\mathbb{Q}[x]/\left\langle x^2-13\right\rangle$, compute the sum $\rho_1+\rho_2$, the product $\rho_1\rho_2$, and the squares $\rho_1^2$ and $\rho_2^2$.
- If you needed to program a computer to store and manipulate the real solutions of the equation $x^2-5x+3=0$, what approach would you take?
