= Solution
The <nematic order parameter> is a <symmetric second-rank tensor> and a <traceless second-rank tensor> describing orientational anisotropy. For molecular unit axes $\mathbf s$, $Q_{ij}$ is proportional to $\langle s_i s_j-\delta_{ij}/3\rangle$. It vanishes in an <isotropic phase> and is unchanged by head-tail reversal $\mathbf s\mapsto-\mathbf s$, unlike a <polar order parameter>. For <uniaxial nematic order>, the <nematic director> is thus an unoriented axis.
One convention for the <Landau-de Gennes free energy> through fourth order, respecting <rotational symmetry>, is
$$
\boxed{f(Q)=\frac a2\operatorname{Tr}Q^2
+\frac c3\operatorname{Tr}Q^3
+\frac b4(\operatorname{Tr}Q^2)^2},\qquad b>0.
$$
For a <traceless second-rank tensor> in three dimensions, the <Cayley-Hamilton theorem> gives $\operatorname{Tr}Q^4=(\operatorname{Tr}Q^2)^2/2$, so there is only one independent quartic <rotational invariant of a symmetric traceless tensor>. The only possible linear scalar is $\operatorname{Tr}Q=0$; without an external anisotropy no linear term survives.
The cubic <rotational invariant of a symmetric traceless tensor> distinguishes prolate and oblate forms of <uniaxial nematic order>. Head-tail reversal leaves $Q$ unchanged and does not impose $Q\mapsto-Q$. Thus a cubic term is allowed in three dimensions and generically produces a <first-order phase transition>, rather than a symmetry-enforced continuous onset. In two dimensions a symmetric <traceless second-rank tensor> has <eigenvalues> $s,-s$, hence $\operatorname{Tr}Q^3=0$: \b[the cubic invariant vanishes identically in two dimensions]. This is a statement about the <Landau free energy>; fluctuations in two dimensions require a separate treatment.
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