Question

IfKis an $\mathit{n}\mathbf{-}$dimensional extension field of ${\mathit{Z}}_{\mathbf{p}}$, what is the maximum possible number of element in K.

Short answer

Answer:

$p^n$is the maximum number in K.

Step-by-step solution

Definition of simple extension.

A simple extension is a field extension which is generated by the adjunction of single element. Every finite field is a simple extension of the prime field of the same characteristic.

Showing that pn is the maximum number in K.

let K is a n-dimensional extension field of ${\mathit{Z}}_{\mathbf{p}}$.

As, $\left[K:Z_p\right]\mathbf{=}\mathit{n}$

So there exists a basic for K over ${\mathit{Z}}_{\mathbf{p}}$that consists of n elements.

Let $\mathit{\beta }\mathbf{=}\left\{v_1.v_2,.....v_n\right\}$be the basic for K over ${\mathit{Z}}_{\mathbf{p}}$that consists of n element.

So any arbitrary element $\mathit{x}\mathbf{\in }\mathit{K}$can be written as,

$\mathit{x}\mathbf{=}{\mathit{\alpha }}_{\mathbf{1}}\mathbf{,}{\mathit{v}}_{\mathbf{1}}\mathbf{+}{\mathit{\alpha }}_{\mathbf{2}}{\mathit{v}}_{\mathbf{2}}\mathbf{+}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}{\mathit{\alpha }}_{\mathbf{n}}{\mathit{v}}_{\mathbf{n}}$where ${\mathit{\alpha }}_{\mathbf{1}}\mathbf{\in }{\mathit{Z}}_{\mathbf{p}}$

This implies each ${\mathit{\alpha }}_{\mathbf{1}}$has p-choices. Therefore possible value for x is ${\mathit{p}}^{\mathbf{n}}$.

Therefore the maximum number of element in K is ${\mathit{p}}^{\mathbf{n}}$.