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Differentiate the Legendre differential equation and put $Q_n=P_n^{\prime }$. This givesMultiplication by $1-x^2$ puts it in the self-adjoint formThe boundary term vanishes at $x=\pm 1$. Distinct eigenvalues are therefore orthogonal with weight $1-x^2$:

At a regular constrained stationary point of $F$ on $G=0$, the Lagrange multiplier method solvesFor the first problem, substitute $z=1+xy$ to obtainIts stationary equations imply either $x=y$ or $xy=0$, and in either case the only real stationary point is $x=y=0$, $z=1$. Coercivity ensures that the minimum is attained, soat $(0,0,1)$.

For the second problem, the arithmetic-geometric mean inequality givesConsequentlyEquality occurs at $(0,0,1)$, and hence

With the stated Fourier transform convention, Fourier inversion givesFor $x>0$, close the contour in the upper half-plane. Jordan lemma removes the semicircle contribution, and the only enclosed pole is $k=ip$. Its residue isThe residue theorem therefore yields

The first isomorphism theorem for rings states that a ring homomorphism $\varphi :R\to S$ induces an isomorphismBecause $J$ is an ideal, sums and products of elements of $R+J$ remain in $R+J$, so it is a subring. The surjective homomorphismhas kernel $R\cap J$. The theorem givesEvaluation at $-2$ givesso this quotient is a field of characteristic zero. The other quotient isIt has characteristic $3$ but is not a field, since the nonzero class of $X-1$ is nilpotent.