Finding the zeroes of a polynomial—values of

Eventually
Paolo Ruffini
and
Niels Abel
were able to
prove
that *there is no quintic formula*. This is somewhat surprising; even though
the zeroes exist,
there is no single finite
expression
of +, -, ×, ÷, and
radicals
that can produce them from the coefficients for all quintics. (One can resort to *infinite* expressions;
Newton's method
provides one. See also
‘limit of a sequence’.)

But their proof did not generalise to higher degrees. The honour of proving the quartic formula to be the last of its kind, ie there was no sextic, septic, octic, formula, and so on, fell to Evariste Galois, who had an ingenious insight which reduced the issue to an important but solved question of group theory.