Which algebraic intuition can be used in fields

Question

I wonder what basic laws of arithmetic of reals e.g. $x^n x^m = x^{m+n}$ holds for fields. Every time I take some book on abstract algebra it proves very abstract and unpractical properties. So I wonder which algebraic intuition I can use when working with some new field of course except field axioms themselves.

Answer 1

Some of the most important properties are that polynomials work similarly.

1. The concept of factoring polynomials retains its connection to finding roots via the Polynomial remainder theorem.

2. A polynomial of degree $n$ therefore still has at most $n$ roots.

3. More generally, the concept of long division with polynomials continues to work, which implies the above and many other things.

4. You can still adjoin imaginary numbers to solve irreducible polynomials (that is, every polynomial splits in some extension of the field), just like you can do to get from $\mathbb R$ to $\mathbb C$.