On page 598 of
MATHEMATICAL THOUGHT FROM ANCIENT TO MODERN TIMES,
Morris Kline writes
The first substantial proof of the fundamental theorem [of algebra],
though not rigorous by modern standards, was given by Gauss in his
doctoral thesis of 1799 at Helmstadt. He criticized the work of
d'Alembert, Euler, and Lagrange and then gave his own proof
.
On the same page, Kline writes
Gauss gave three more proofs.
However, on page 224 of Modern Algebra, English translation 1949, 1953,
B. L. Van der Waerden writes
Gauss gave five proofs of the fundamental theorem.
My questions:
Did Gauss give four proofs or five?
Are any proofs by Gauss "rigorous by modern standards"?
The easiest proof & the only one that I have ever studied, follows
from Liouville's theorem, and if I remember correctly, it is steeped in
analysis. Is it possible to prove the fundamental theorem of
algebra strictly algebraically, with no element of analysis?
My planner from last January indicates that I have seen a book with the
title
The Fundamental Theorem of Algebra
by Fine and Rosenberger. Does anyone on the list know the merits of
this text?
Best wishes from Annapolis,
Sam Kutler