I need to construct any new type of Construct-able field for my purpose of research purpose in geometry. But I have doubt whether it is easy to found or most tougher problem.
Can I construct or find out a new kind of field ?
Answer:
I am thinking that to construct a new type of any field surely difficult .
Can I manipulate a new kind of field from our well-known field $ \ \mathbb{R} \ \ and \ \ \mathbb{Q} \ $ by imposing extra elements or extra elements .
But I am getting no inspiration because I am thinking it will be in vain in advanced.
Therefore I need your valuable hints or advice or answer.
Help me with at least a little conclusion .
For a field, two important aspects: cardinality and characteristic.
Finite fields or Galois fields with symbol $\mathbb{F}_{p^n}$ are constructed and studied using quotients of polynomials and so on.
For fields with characteristic zero, with all of them contain $\mathbb{Q}$, we have some famous fields like $\mathbb{R}$ and $\mathbb{Q}_p$ (p-adic numbers). These two fields are constructed using concept of metric and completion of rationals $\mathbb{Q}$ and there is only this two type of fields which can be constructed using "completion" method.
Also, field of complex numbers $\mathbb{C}$, skew field of quaternions, $\mathbb{H}$ and octonions $\mathbb{O}$ and so on, constructed using "Cayley-Dickson" method, which constructs $2^n$-dimensional $\mathbb{R}$-algebras over Cartesian product $\mathbb{R}\times\cdots\times\mathbb{R}$.
So if any of these method don't satisfy you, then you need to create your own method to constructing new fields.