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people:fer:402ws:spring2019:homework

## Math 402 - 01 Homework (Spring 2019)


Problem Set 05 (complete) Due: 03/25/2019 Board presentation: 03/??/2019

1. Let $F$ be a field and $f(x), g(x)\in F[x]$. Prove:
1. $(f(x)+g(x))' = f'(x) + g'(x)$
2. $(f(x)g(x))' = f(x)g'(x) + f'(x)g(x)$
2. Let $F$ be a field, and $\varphi:F\to F$ an endomorphism of $F$. Prove that the set $F_\varphi=\{a\in F\mid\varphi(a)=a\}$ is a subfield of $F$.
3. How many monic irreducible polynomials of degree 4 are there over $\F_5$?
4. Let $E$ be a field extension of $F$. Prove that $E$ is an algebraic closure of $F$ iff $E$ is minimal with the property that every polynomial $f(x)\in F[x]$ splits over $E$.

Problem Set 04 (complete) Due: 03/11/2019 Board presentatiion: 03/25/2019

1. Let $E/F$ be a field extension. Prove that $[E:F]=1$ iff $E=F$.
2. Let $E$ and $K$ be field extensions of $F$ and $\varphi:E\to K$ an $F$-extension homomorphism. Show that $\varphi$ is a linear transformation of $F$-vector spaces.
3. Write $\sq{2}$ as a polynomial expression on $\alpha=\sq{2}+\sq{3}$.
4. Find the minimal polynomial of $u=(\sq[3]{2}+\omega)$ over $\Q$.
people/fer/402ws/spring2019/homework.txt · Last modified: 2019/03/16 13:57 by fer