congruence of arbitrary degree
We suppose now that the congruence
where , has at least incongruent roots . Form the congruence
Both sides have the same term of the highest degree, whence they may be cancelled from the congruence and the degree of (2) has a lower degree than . Because (2), however, clearly has incongruent roots , it must by the induction hypothesis be simplifiable to the form and thus be an identical congruence.
Now, if the congruence (1) had an additional incongruent root , i.e. , then the identical congruence (2) would imply
Example. When , we have
Thus only the representants 2 and 4 of a complete residue system modulo 7 (see conditional congruences) are roots of the given congruense. A congruence needs not have the maximal amount of incongruent roots mentionned in the theorem.
- 1 K. Väisälä: Lukuteorian ja korkeamman algebran alkeet. Tiedekirjasto No. 17. Kustannusosakeyhtiö Otava, Helsinki (1950).
|Title||congruence of arbitrary degree|
|Date of creation||2013-03-22 18:52:29|
|Last modified on||2013-03-22 18:52:29|
|Last modified by||pahio (2872)|