# Which algebraic intuition can be used in fields

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.

• $x^ny^m\neq (xy)^{m+n}$ in general, even for the real numbers. Jan 31, 2015 at 20:52
• I guess it might be a better question asking, what holds for the reals but does not hold in a field in general Jan 31, 2015 at 20:53
• @mvw what about $\mathbb{Q}$? Jan 31, 2015 at 20:58
• @mvw you must be thinking of the result that the reals are the largest Archimedian ordered field. Jan 31, 2015 at 21:15
• The idea was to see what extra properties in addition to being a field characterize the reals or rationals etc and then think which calculation rules have to be dropped without those.
– mvw
Jan 31, 2015 at 21:27

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