1. Finite Field

Prove the following: Every finite field is perfect.

Proof. Suppose $\displaystyle F$ is a finite field of characteristic $\displaystyle p$ and let $\displaystyle E$ be a finite extension of $\displaystyle F$. Also let $\displaystyle \alpha \in E$. We want to show that $\displaystyle \alpha$ is separable over $\displaystyle F$. We know that $\displaystyle f(x) = \text{irr}(\alpha, F)$ factors in $\displaystyle \overline{F}$ as $\displaystyle \prod_{i} (x-\alpha_i)^{v}$ where the $\displaystyle \alpha_i$ are the zeros of $\displaystyle f(x)$. We need to show that $\displaystyle v=1$.

What would be a good way to approach this?

2. Originally Posted by Sampras
Prove the following: Every finite field is perfect.

Proof. Suppose $\displaystyle F$ is a finite field of characteristic $\displaystyle p$ and let $\displaystyle E$ be a finite extension of $\displaystyle F$. Also let $\displaystyle \alpha \in E$. We want to show that $\displaystyle \alpha$ is separable over $\displaystyle F$. We know that $\displaystyle f(x) = \text{irr}(\alpha, F)$ factors in $\displaystyle \overline{F}$ as $\displaystyle \prod_{i} (x-\alpha_i)^{v}$ where the $\displaystyle \alpha_i$ are the zeros of $\displaystyle f(x)$. We need to show that $\displaystyle v=1$.

What would be a good way to approach this?

This question, the one on the primitive element and perhaps others are very basic, standard stuff...I mean, are you trying to cope with this material without the aid of books? Any decent book in algebra/fields/extensions/Galois Theory (with introduction on extensions) deal with this (for example, you could first prove that a field K of characteristic p is perfect iff $\displaystyle K^p=K\Longleftrightarrow$ the p-th power map is an automorphism of K.)
I propose you grab some books, try to get some help from them AND THEN, if you get stuck somewhere, you ask.
I firmly believe that anyone studying mathematics at any level above high school (and also before is highly recommended) must get used to books and must have several by his/her side all the time, either of his property or the library's.

Tonio

3. Originally Posted by tonio
This question, the one on the primitive element and perhaps others are very basic, standard stuff...I mean, are you trying to cope with this material without the aid of books? Any decent book in algebra/fields/extensions/Galois Theory (with introduction on extensions) deal with this (for example, you could first prove that a field K of characteristic p is perfect iff $\displaystyle K^p=K\Longleftrightarrow$ the p-th power map is an automorphism of K.)
I propose you grab some books, try to get some help from them AND THEN, if you get stuck somewhere, you ask.
I firmly believe that anyone studying mathematics at any level above high school (and also before is highly recommended) must get used to books and must have several by his/her side all the time, either of his property or the library's.

Tonio
Do you think using too many books destroys intuition? E.g. what about "discovery-based learning?"

4. Originally Posted by Sampras
Do you think using too many books destroys intuition? E.g. what about "discovery-based learning?"

I'm not sure I fully understand what you mean, but if I get it right the yes: if you ALWAYS manage to find an answer to your questions in books and/or in some maths forum then not precisely your intuition but rather your creativity and/or your working habits can get impaired.

Yet, in the case of standard, non-trivial well-known theorems, I think it is a GOOD habit to have reference books by your side, work by yourself proofs and, in case of getting stuck, then ask someone else.

Tonio