Abel's theorem in problems and solutions - School of Mathematics

72 Chapter 2
265. Let Prove that
266. Let Prove that
Let us denote by the curve with equation
(i.e., the circle with radius equal to R, oriented counterclockwise).
Since the curve is closed the curve
is closed as well Let be the
number of turns of the curve around the point (if
does not pass through the point
267. Calculate and
Let us now increase the radius R from to The curve
will consequently be deformed from to If for a value R*
the curve does not pass through the point by a sufficiently
small variation of R near R* the curve will turn out to be deformed
by too small an amount for the number of its turns around the point
to change: the function is indeed continuous at the value R*. If the
curves avoid the point for all values of R between and
then is a continuous function for all Since the
function takes only integer values it can be continuous only if for
all values of it takes a unique value. But, solving Problem
267, we have obtained that and Therefore the
claim that none of the curves passes through the point for
all is untrue. We thus have for a certain In
this way we have proved the following theorem 8 .
THEOREM 7. (The fundamental theorem of the algebra of complex
numbers 9 ). The equation
8 Our reasoning contains some non-rigourous passages: it must be considered, in
general, as an idea of the proof. This reasoning can, however, be made exact (though
in a non-simple way). See, for example, Chinn W.G., Steenrod N.E, (1966), First
Concepts of Topology, (Mathematical Association of America: Washington).
9 This theorem was proved in 1799 by the German mathematician C.F. Gauss (1777–
1855).