Help with Dummit & Foote Exercise 1, Section 13.2 - Algebraic Extensions

[Math Amateur]

I need help with Exercise 1 of Dummit and Foote, Section 13.2 : Algebraic Extensions ..

I have been unable to make a meaningful start on the problem ... ... Exercise 1 of Dummit and Foote, Section 13.2 reads as follows:
View attachment 6608
I have been unable to make a meaningful start on this problem ...BUT ... further I hope to understand why D&F put this exercise at the end of a section on algebraic extensions ... the exercise seems a bit remote from the subject matter ... ...
Hope someone can help ...Peter

 

[HallsofIvy]

I would start with the definition of "characteristic" of a field. What is that definition?

 

[Math Amateur]

I would start with the definition of "characteristic" of a field. What is that definition?

Thanks HallsofIvy ...

Thought about definition and its link to the prime subfield ... and had considerable help from people on the Physics Forums ...Have the following solution ... please critique it ...============================================================================

\(\displaystyle F\) finite field of characteristic \(\displaystyle p\)\(\displaystyle \Longrightarrow\) prime subfield of \(\displaystyle F\) is isomorphic to \(\displaystyle \mathbb{F}_p\) and \(\displaystyle p\) must be prime ... (Lidl and Niederreiter, Introduction to Finite Fields ... ... , Theorem 1.78 ... ...also \(\displaystyle F\) finite means that \(\displaystyle F\) has finite dimension over \(\displaystyle \mathbb{F}_p\), say dimension of \(\displaystyle F\) over \(\displaystyle \mathbb{F}_p = n\) \(\displaystyle \Longrightarrow\) Basis for \(\displaystyle F\) over \(\displaystyle \mathbb{F}_p\) has \(\displaystyle n\) elements\(\displaystyle \Longrightarrow\) all elements of \(\displaystyle F\) are uniquely expressible as \(\displaystyle c_1 v_1 + c_2 v_2 + \ ... \ ... \ \ c_n v_n\) ... ... ... (1)where \(\displaystyle c_1, c_2, \ ... \ ... \ \ , c_n \in \mathbb{F}_p\)

and

\(\displaystyle v_1, v_2, \ ... \ ... \ \ , v_n \in F\)
\(\displaystyle \Longrightarrow\) number of elements in \(\displaystyle F, |F| = p^n\)

since each \(\displaystyle c_i\) in expression (1) has \(\displaystyle p\) possibilities ... ... ( and the \(\displaystyle v_i\) are fixed)=========================================================================Is the above correct?Could someone please critique solution/proof ... is it rigorous ...?Peter