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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.

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    $\begingroup$ $x^ny^m\neq (xy)^{m+n}$ in general, even for the real numbers. $\endgroup$ Jan 31, 2015 at 20:52
  • $\begingroup$ I guess it might be a better question asking, what holds for the reals but does not hold in a field in general $\endgroup$ Jan 31, 2015 at 20:53
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    $\begingroup$ @mvw what about $\mathbb{Q}$? $\endgroup$ Jan 31, 2015 at 20:58
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    $\begingroup$ @mvw you must be thinking of the result that the reals are the largest Archimedian ordered field. $\endgroup$
    – rschwieb
    Jan 31, 2015 at 21:15
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    $\begingroup$ 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. $\endgroup$
    – mvw
    Jan 31, 2015 at 21:27

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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 $\mathbb C$.

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