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For a constant-coefficient linear recurrence relation over the complex numbers, a nonzero root $\mu$ of multiplicity $m$ in its characteristic polynomial of a linear recurrence supplies the $m$ linearly independent solutionsSubstitute $y_n={\mu }^n{z}_n$. Each factor $E-\mu$ of the shift linear operator becomes $\mu$ times the forward difference operator on $z_n$, so its $m$th power kills every polynomial in $n$ of degree less than $m$. Their independence follows from the Lagrange root bound over a field. The same argument works over a field of characteristic zero; the displayed monomials need modification in positive characteristic.

A zero root needs separate initial-data treatment: for characteristic polynomial ${\lambda }^m$, the recurrence relation is $y_{n+m}=0$, leaving the first $m$ values arbitrary.

Write $f(x)=a_m{\prod }_{i=1}^m(x-{\alpha }_i)$ and $g(x)=b_n{\prod }_{j=1}^n(x-{\beta }_j)$ in an algebraic closure, counting multiplicity of a root. The coefficient-row convention above givesThis makes the shared-root of a polynomial test transparent: a factor vanishes precisely when one ${\alpha }_i$ equals one ${\beta }_j$. Swapping the two polynomials reverses the sign of all $mn$ differences, giving $\mathrm{Res}(g,f)=(-1{)}^{mn}\mathrm{Res}(f,g)$.^[ref]

The value measures an algebraic product, not a normalized geometric distance. Replacing $f$ by $cf$ for nonzero $c\in K$ leaves its roots of a polynomial unchanged but multiplies the resultant by $c^n$. Therefore a small numerical resultant alone does not quantify how close two sets of roots of a polynomial are. Exact zero testing and numerical sensitivity are different questions.

A multiple root of a polynomial is also a root of a polynomial of its formal derivative. In characteristic zero, for $\mathrm{deg}f=m\ge 2$, the associated discriminant isFor example, $h(x)=x^2+s$ gives $\mathrm{Res}(h,2x)=4s$ and $\mathrm{Disc}(h)=-4s$. The two roots of a polynomial merge precisely at $s=0$.^[ref]

In characteristic zero, averaging over a compact group, or over a Zariski-dense compact form of a complex reductive group, gives a degree-preserving projection onto the polynomial invariant ring. Its module property is $\mathcal{R}(af)=a\mathcal{R}(f)$ for invariant $a$. This property turns finite ideal generation in the ambient polynomial ring into finite algebra generation of the invariant subring.

If an irreducible quartic over a field of characteristic zero has a cubic resolvent of a quartic whose Galois group of a polynomial is $S_3$, then the quartic has Galois group $S_4$. Irreducibility places the quartic group among the transitive subgroups of the symmetric group on four points, while the action on its three root pairings is surjective onto $S_3$. Of the possibilities $C_4,V_4,D_8,A_4,S_4$, only $S_4$ has this image.

A transcendence basis $T$ for $L/k$ is separating if $L/k(T)$ is a separable field extension. In this case the $dt$, $t\in T$, form a basis of the Kähler differentials. In characteristic zero every transcendence basis is separating. In positive characteristic the condition on the chosen basis is essential.

In a nonzero two-sided ideal choose an element of least Bernstein filtration degree. Commuting it with any position or differentiation generator lowers degree, so all such commutators vanish. The ordered monomial basis of a Weyl algebra and characteristic zero then force it to be a nonzero scalar. The ideal is the whole ring. The argument fails in positive characteristic, when powers of the generators can be central.

Over a field $k$, the trace-zero $2\times 2$ matrices form the special linear Lie algebra ${\mathfrak{sl}}_2(k)$. The matrix trace of a commutator is zero, so this is a Lie subalgebra of the general linear Lie algebra. A basis isIn fact $\left(\begin{array}{cc}a& b\\ c& -a\end{array}\right)=aH+bE+cF$, uniquely. The individual products areSubtracting in the stated order givesFor characteristic different from two, this is the standard sl2 triple; see Etingof, Section 11.4. The products remain valid in characteristic two, but there $H=I$ is central and $2E=2F=0$, so the familiar characteristic zero structure changes.

The following table gives all nine basis brackets. Each entry is $[\text{row},\text{column}]$; the diagonal vanishes by alternation, and exchanging row and column reverses the sign.

Bilinearity now computes any commutator, for exampleThe three structure constants of a Lie algebra relations encode all of this multiplication. They also show why the sl2 triple is useful in Lie algebra representations: the actions of $E$ and $F$ move between eigenvalues of the action of $H$ in steps of two.

Over a field of characteristic zero, an sl2 triple consists of elements of a Lie algebra satisfying the displayed brackets. It defines a homomorphism from ${\mathfrak{sl}}_2$ by sending its standard generators to $h,e,f$. A nonzero homomorphism is injective because ${\mathfrak{sl}}_2$ is a simple Lie algebra. Root vectors and their coroot give such triples in a semisimple Lie algebra, and their representations supply finite weight strings.

Over a field of characteristic zero, a polynomial is solvable by radicals when its roots lie in a tower obtained by adjoining successive roots; equivalently, its Galois group is solvable.

For a finite-dimensional Lie algebra over a field of characteristic zero, the radical ${\mathfrak{g}}^{\perp }=\{x:\kappa (x,\mathfrak{g})=0\}$ of the Killing form is a solvable ideal. Invariance makes it an ideal, and the Cartan solvability criterion applied to its adjoint image proves solvability.

A finite separable extension is solvable when its normal closure has a solvable group as Galois group. In characteristic zero it is contained in a radical tower after adjoining suitable roots of unity; containment need not mean it is itself a tower of radical adjunctions.

Over a field $k$, the special linear Lie algebra is ${\mathfrak{sl}}_n(k)=\{X\in M_n(k):\mathrm{tr}X=0\}$ with Lie bracket $[X,Y]=XY-YX$. The trace of a matrix commutator vanishes, so this vector subspace of the general linear Lie algebra is a Lie subalgebra. For $n\ge 1$ its dimension is $n^2-1$.

For $n=2$, the matrix units $e=E_{12}$, $f=E_{21}$ and $h=E_{11}-E_{22}$ give the sl2 Lie algebra relations $[h,e]=2e$, $[h,f]=-2f$ and $[e,f]=h$. In characteristic zero, ${\mathfrak{sl}}_n$ is a simple Lie algebra for $n\ge 2$. In characteristic $p$ dividing $n$, the identity matrix has trace zero and spans a nonzero central ideal, so that simplicity assertion fails. Over the real numbers, differentiating the determinant condition on $S{L}_n$ identifies this with the Lie algebra of a matrix Lie group of the special linear group.

A semisimple Lie algebra over a field of characteristic zero is split if it has a Cartan subalgebra whose adjoint action is simultaneously diagonalizable over that field. Its root-space decomposition then needs no extension of scalars. The sl2R Lie algebra is split, with the real diagonal Cartan subalgebra. The real special unitary Lie algebra $\mathfrak{su}(2)$ is not: its adjoint operators are skew-adjoint for the positive definite inner product obtained by negating its Killing form. Thus a split adjoint operator would have to be zero.

Here $\mathcal{O}$ is a complete discrete valuation ring with fraction field $K$ of characteristic zero and residue field $k$ of characteristic $p$. The system is splitting for the finite group and its subgroups when their simple representations over $K$ and $k$ are absolutely irreducible. It permits ordinary and modular representation theory to be compared by reduction of lattices modulo the maximal ideal.

The nth symmetric power $S^{n}V$ is the quotient of $V^{\otimes n}$ that identifies tensors differing by a permutation of their factors. Equivalently, over characteristic zero it is the subspace of symmetric tensors.

Let $\mathfrak{a}$ be a toral subalgebra and an abelian Lie algebra in the symplectic Lie algebra, over an algebraically closed field of characteristic zero. Its defining representation has a weight decomposition. The symplectic Lie algebra identity gives $(\lambda (a)+\mu (a))\omega (v,w)=0$ for $v\in V_{\lambda }$, $w\in V_{\mu }$, so $\omega (V_{\lambda },V_{\mu })=0$ unless $\mu =-\lambda$. Since $\omega$ is a nondegenerate bilinear form, its restrictions give a perfect pairing between $V_{\lambda }$ and $V_{-\lambda }$, and a nondegenerate bilinear form on $V_0$.

An element of the centralizer of a Lie subalgebra preserves each weight space. On $V_0$ it is any element of $\mathfrak{sp}(V_0)$. On each nonzero pair, its action $A$ on $V_{\lambda }$ is arbitrary, and the perfect pairing uniquely forces the action on $V_{-\lambda }$ to be the negative dual map $-A^{\ast }$. These independent choices preserve Lie brackets, proving the displayed direct sum of Lie algebras; the sum runs over one representative per nonzero pair.

This centralizer of a Lie subalgebra is an abelian Lie algebra precisely when $V_0=0$ and all the nonzero weight spaces are one-dimensional. Indeed, ${\mathfrak{gl}}_m$ is nonabelian for $m\ge 2$ by $[E_{11},E_{12}]=E_{12}$, while every nonzero symplectic vector space contains a two-dimensional symplectic subspace on which $[\mathrm{diag}(1,-1),E_{12}]=2E_{12}$. In the stated case the surviving summands are copies of ${\mathfrak{gl}}_1$ and commute. Repeated weights of a representation, including a zero one, are therefore exactly what obstructs this commutativity.

A toral subalgebra of a Lie algebra $\mathfrak{g}$ is a Lie subalgebra $\mathfrak{a}$ such that $\mathrm{ad}a$ is diagonalizable for every $a\in \mathfrak{a}$. Here diagonalizability is over the stated ground field, and refers to the Adjoint representation. In arguments that use a simultaneous weight decomposition, commutativity of $\mathfrak{a}$ is stated as well. For a semisimple Lie algebra over an algebraically closed field of characteristic zero, Cartan subalgebras are examples. Semisimplicity on the defining representation of a symplectic Lie algebra follows from adjoint semisimplicity in the symplectic Lie algebra.

The truncated tensor algebra discards products of degree greater than $N$. Its multiplication is $(a\otimes b{)}^{(k)}={\sum }_{j=0}^k{a}^{(j)}\otimes b^{(k-j)}$. Every element with scalar part $1$ is invertible by a finite geometric series in its positive-degree part. Over a field of characteristic zero, truncated exponential and logarithm are finite polynomials. More generally, their coefficient formulas require the relevant denominators through degree $N$ to be invertible in the coefficient ring.

In the operator sense, a finite-dimensional group representation is unipotent when every represented element is a unipotent matrix: for each $g$, $\rho (g)-I$ is a nilpotent endomorphism. This terminology is distinct from the special unipotent characters in the representation theory of finite groups of Lie type.

For example, the generator of $\mathbb{Z}$ can act on a two-dimensional vector space by $T=\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$. Then $T^n=\left(\begin{array}{cc}1& n\\ 0& 1\end{array}\right)$, and the first coordinate line gives a nonsplit extension of representations of the trivial representation by itself. Over a field of characteristic zero, a unipotent matrix of finite order is the identity, because its minimal polynomial divides both a power of $X-1$ and the separable polynomial $X^m-1$. Thus every unipotent group representation of a finite group is trivial in characteristic zero; in characteristic $p$, the displayed $T$ instead gives a nontrivial group representation of $C_p$.

The first Weyl algebra over a field $k$ isIn characteristic zero, it acts faithfully on $k[x]$: $x$ multiplies by the variable and $\partial$ differentiates. The defining relation is the product rule, since $\partial (xf)-x\partial (f)=f$. Every element has a unique ordered expression ${\sum }_{i,j}{a}_{ij}{x}^i{\partial }^j$, giving the ordered monomial basis of a Weyl algebra.

The higher Weyl algebra $A_n(k)$ has commuting coordinates $x_i$, commuting derivatives ${\partial }_i$, and $[{\partial }_i,x_j]={\delta }_{ij}$. It organizes polynomial-coefficient differential operators. The abstract algebra remains meaningful in positive characteristic, but its usual action on polynomials is no longer faithful: ${\partial }^p$ acts as zero in characteristic $p$ while remaining a nonzero ordered monomial. See Weyl algebra in positive characteristic for the resulting finite-dimensional representations.