1. ## (X^4)+(4x^2)+144

for the following equation in the title, can anyone help us to find out the following information:
1) how can this equation be used to contruct a field?
2)how does this connect with algerbraic elements?
3) with maximal ideals?
4)with contructable real numbers?

2. Originally Posted by vpatel
for the following equation in the title, can anyone help us to find out the following information:
1) how can this equation be used to contruct a field?
2)how does this connect with algerbraic elements?
3) with maximal ideals?
4)with contructable real numbers?

For example, find a prime finite field $\displaystyle \mathbb{F}_p:=\mathbb{Z}/p\mathbb{Z}$ where this pol. is irreducible , and

then the ideal $\displaystyle M:=\langle x^4+4x^2+144\rangle$ is maximal in $\displaystyle \mathbb{F}_p[x]\Longrightarrow \mathbb{F}_p[x]/M$ is a field (with $\displaystyle p^4$ elements)...

Of course, you can try the above also with some field $\displaystyle \mathbb{F}$ with characteristic zero...

Tonio