Gauss’s lemma II
Remark. The above statement is often used in its contrapositive form. For an example of this usage, see this entry (http://planetmath.org/AlternativeProofThatSqrt2IsIrrational).
Proof. A primitive polynomial in is by definition divisible by a non invertible constant polynomial, and therefore reducible in (unless it is itself constant). There is therefore nothing to prove unless (which is not constant) is primitive. By assumption there exist non-constant such that . There are elements such that and are in and are primitive (first multiply by a nonzero element of to chase any denominators, then divide by the gcd of the resulting coefficients in ). Then is primitive by Gauss’s lemma I, but is primitive as well, so is a unit of and is a nontrivial decomposition of in . This completes the proof.
Remark. Another result with the same name is Gauss’ lemma on quadratic residues.
From the above proposition and its proof one may infer the
|Title||Gauss’s lemma II|
|Date of creation||2013-03-22 13:07:52|
|Last modified on||2013-03-22 13:07:52|
|Last modified by||bshanks (153)|
|Synonym||Gauss’ lemma II|