MHB Zach's question at Yahoo Answers (Field with 25 elements)

[Fernando Revilla]

Here is the question:

a) Construct a field of order 25

b) Describe the elements of Q(sqrt5)

Here is a link to the question:

Abstract Linear Algebra? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.

 

[Fernando Revilla]

Hello Zach,

$(a)$ The field $\mathbb{Z}_5$ has 5 elements and consider $p(x) = x^2 + 2\in\mathbb{Z}_5[x]$. This polynomial has no zeroes in $\mathbb{Z}_5$ and a quadratic polynomial without zeroes is irreducible. Hence, $F=\mathbb{Z}_5[x]\;/<x^2+2>$ is a field. But every class has one and only one representative of the form $ax+b$ with $a,b\in\mathbb{Z}_5 $. This implies $\#(F)=5\cdot 5=25.$

$(b)$ According to the theory of field extensions, the minimal polynomial of $\sqrt{5}$ is $f(x)=x^2-5$, so a basis of $[\mathbb{Q}(\sqrt{5}):\mathbb{Q}]$ is $B=\{1,\sqrt{5}\}$. As a consequence, $\mathbb{Q}(\sqrt{5})=\{a+b\sqrt{5}:a,b\in\mathbb{Q}\}$.