Newton’s identities give rearrangement formulae for the elementary symmetric variety of these functions. For 10 points each:
[10e] Name these functions expressed as sums of powers of a variable. Examples of these functions of degree 2 may be solved with the quadratic formula.
ANSWER: polynomials
[10m] By Gauss’s lemma, a primitive polynomial is irreducible over the integers if and only if it is irreducible over these numbers. A namesake theorem constrains roots in this set in terms of a polynomial’s first and last coefficients.
ANSWER: rational numbers [or rationals; accept Q]
[10h] Another sufficient criterion for the irreducibility of polynomials over the rational numbers is given by this German mathematician. Gauss apocryphally rated this mathematician on par with Newton and Archimedes.
ANSWER: Gotthold Eisenstein [or Ferdinand Gotthold Max Eisenstein]
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