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A Lie bracket is alternating when $[x,x]=0$ for every $x$. This condition is imposed in every characteristic of a field. Bilinearity gives $0=[x+y,x+y]=[x,y]+[y,x]$, hence antisymmetry of a Lie bracket. Requiring alternation directly keeps the definition correct in characteristic two.

A vector is anisotropic for a quadratic form if $Q(v)\ne 0$, and for a Hermitian form if $h(v,v)\ne 0$. The complementary zero-value condition is isotropy. For a symmetric bilinear form in odd characteristic of a field, this agrees with the associated quadratic form. A nondegenerate alternating bilinear form has all self-pairings zero, so nondegeneracy must not be confused with the existence of anisotropic vectors.

The commutator of two elements of an associative algebra over a field isIt measures how much the order of multiplication matters: $[A,B]=0$ exactly when $AB=BA$. It is a bilinear map, it is alternating since $AA-AA=0$, and associativity makes it satisfy the Jacobi identity. Hence every associative algebra becomes a Lie algebra on the same vector space when equipped with this product. This holds in every characteristic of a field.

For square matrices, this is the matrix commutator, and it is defined even for singular matrices. The group commutator of invertible matrices is instead $AB{A}^{-1}{B}^{-1}$: it equals the identity element when the matrices commute, whereas $AB-BA$ equals the zero matrix. The Lie bracket from local group commutators explains how differentiation connects the two constructions.

Let $S$ send successive standard basis vectors of $K^n$ to the next one, wrapping the last to the first. Its minimal polynomial is $X^n-1$: this polynomial annihilates it, and the cyclic vectors $e_1,S{e}_1,\dots ,S^{n-1}{e}_1$ form a basis, excluding any annihilator of smaller degree. The diagonalizability criterion using the minimal polynomial gives the displayed equivalence. A splitting polynomial is square-free exactly when the characteristic of a field does not divide $n$, as follows by taking its derivative $n{X}^{n-1}$ and checking common roots. Splitting over the specified field is a separate requirement. For $n=3$, the shift is diagonalizable over the complex numbers and ${\mathbb{F}}_7$, but not over ${\mathbb{F}}_2$ or ${\mathbb{F}}_3$.

A division point above $P$ is a point $Q$ over an algebraic closure satisfying $[N]Q=P$, for a specified nonzero integer $N$. Two such points differ by an $N$-torsion point of an elliptic curve. If the characteristic of a field does not divide $N$, there are $N^2$ distinct division points; in other characteristics the scheme fibre still has length $N^2$, but its geometric points can be fewer. The unramified division torsors at good primes control their coordinate fields over local fields.

A Lie algebra representation on $V$ induces one on each exterior power by acting on every factor and summing. The tensor product of Lie algebra representations preserves the defining alternating relations, so this descends to the quotient in every characteristic of a field. In characteristic two define the exterior algebra by the relations $v\wedge v=0$, rather than by dividing a tensor antisymmetrizer by $2$.

A finite field is a field with finitely many elements: addition, subtraction, multiplication, and division by a nonzero element all stay inside a finite collection. Write ${\mathbb{F}}_q$, or $\mathrm{GF}(q)$, for a finite field with $q$ elements. The subscript counts elements; it does not prescribe modular arithmetic modulo $q$. The field with four elements is the smallest example where that distinction matters.

Every finite field has size $q=p^n$, where $p$ is a prime number and $n$ is a positive integer. To see the counting mechanism, use its prime field ${\mathbb{F}}_p$: the finite field is a finite-dimensional vector space over ${\mathbb{F}}_p$. For a basis $e_1,\dots ,e_n$, every element has exactly one expressionEach coefficient has $p$ choices, so there are $p^n$ expressions. Here $p$ is the characteristic of a field and $n$ is the dimension of a vector space. Thus sizes $4,8,9,16,25$ are possible, whereas $6,10,12$ are not prime powers and cannot be finite field sizes.

Conversely, every prime power occurs, and any two finite fields of the same size admit a field isomorphism. These are classification statements, beyond the counting argument. See Conrad's Theorems 1.5, 2.2, and 2.7.^[ref] Uniqueness means the arithmetic agrees after relabeling; it does not choose a basis, an irreducible polynomial for a quotient ring representation, or a particular field isomorphism. Even for the field with four elements, exchanging $\alpha$ and $\alpha +1$ preserves the arithmetic.

The appeal for coding theory is exact, finite arithmetic with reliable division: polynomial interpolation and linear algebra work over a finite alphabet. A Reed-Solomon code represents messages by polynomials and adds redundancy by evaluating them at several distinct finite field elements. Forney's MIT notes develop this arithmetic as preparation for such error-correcting codes.^[ref] In cryptography, finite fields likewise supply exact arithmetic on finitely many symbols, with operations that can be implemented using bits.

Historically, Évariste Galois published work on finite fields beyond prime fields in 1830, constructing them by adjoining elements satisfying irreducible polynomials over prime fields. Conrad's historical account also records E. H. Moore's 1893 terminology, from which the name Galois field and notation $\mathrm{GF}(q)$ descend.^[ref]

In characteristic $p$, the coefficient $i$ in a formal derivative vanishes exactly when $p$ divides $i$.

For a prime number $p$, a nonprincipal ultrafilter on $\mathbb{N}$ givesThe Łoś theorem makes $K$ a field of characteristic $p$. For every natural number $m$, almost all factors have at least $m$ elements, so $K$ is infinite.

For a Weierstrass equation of an elliptic curve, use the local parameter $t=-x/y$ and put $w=-1/y$. The formal kernel of a minimal Weierstrass equation construction gives $w=t^3+a_1{t}^4+(a_1^2+a_2)t^5+(a_1^3+2a_1{a}_2+a_3)t^6+O(t^7)$. Substitution into the elliptic curve group law yields the displayed initial coefficients and the formal inverse $-t/(1-a_{1}t-a_{3}w(t))$. In characteristic zero, the invariant differential on an elliptic curve becomes $(1+a_{1}t+(a_1^2+a_2)t^2+O(t^3))dt$, giving the formal logarithm $t+a_1{t}^2/2+(a_1^2+a_2)t^3/3+O(t^4)$. The addition coefficients are integral polynomial expressions and hence also apply in positive characteristic of a field.

The Jacobi identity is the compatibility condition on a Lie bracketTogether with an Alternating Lie bracket, it defines a Lie algebra. Unlike associativity, it relates three cyclic bracketings rather than equating two bracketings. Equivalently, using antisymmetry of a Lie bracket,Thus the linear map ${\mathrm{ad}}_x:y\mapsto [x,y]$ is a derivation of a Lie algebra: acting on a bracket is the sum of acting on each argument. This makes infinitesimal symmetries act consistently on each other's products. It also gives $[{\mathrm{ad}}_x,{\mathrm{ad}}_y]={\mathrm{ad}}_{[x,y]}$, the identity that makes the Adjoint representation of a Lie algebra a Lie algebra representation.

For the commutator in an associative algebra, the cancellation is directly visible:Each word occurs once with each sign. This proves the Jacobi identity without a restriction on the characteristic of a field.

For a computed check in the sl2 matrix commutator example,Here the last two terms are individually nonzero over characteristic zero; the identity records their cancellation. In the cross-product Lie algebra, the same cyclic cancellation is the vector triple product identity.

The historical connection is unusually concrete: Lie and Engel's volume I expands nested commutators of differential operators into twelve terms and cancels them cyclically, in Chapter 5, §26, equation (7). See Lie and Engel, §26, in Merker's translation, pp. 105–106.

A Lie algebra over a field $k$ is a vector space $\mathfrak{g}$ with a bilinear map $[,]:\mathfrak{g}\times \mathfrak{g}\to \mathfrak{g}$, called the Lie bracket, that is alternating and satisfies the Jacobi identity. Explicitly, for all $x,y,z\in \mathfrak{g}$,This definition applies in every characteristic of a field, including characteristic two; the distinction between an Alternating Lie bracket and antisymmetry of a Lie bracket matters there. See Pavel Etingof, Lie Groups and Lie Algebras, Definition 8.4 and Remark 8.5.

The motivating question is how to describe very small symmetries. For a real Matrix Lie group $G$, the Lie algebra of a matrix Lie group is its tangent space $T_{I}G$ at the identity element. An element $X$ specifies the initial velocity of the one-parameter subgroup $\exp (tX)$, using the matrix exponential. The Lie bracket records the first nonzero effect of changing the order of two small transformations. Even when first-order motions cancel, a second-order motion can remain: the small rotation commutator below leaves a rotation about a third axis.

This turns local symmetry questions into computations with linear maps. A basis and its structure constants of a Lie algebra determine every Lie bracket by bilinearity. For example, three matrices suffice for the sl2 matrix commutator example, while the familiar cross product describes the SO(3) Lie algebra. These descriptions help calculate rotations in three dimensions and build Lie algebra representations; the same bracket structure underlies angular momentum commutation relations. The striking feature is that a nonlinear family of motions has a local description by a vector space and one product. Global information still needs care: the Lie algebra alone does not distinguish the circle group from the additive real line.

Historically, the connection with differential equations is visible in Sophus Lie's Theorie der Transformationsgruppen, volume I (1888), prepared with Friedrich Engel. Chapter 6 interprets first-order differential operators as infinitesimal transformations, linking their action on functions to the transformations those functions admit. See Lie and Engel, Chapter 6, §27, in Joël Merker's English translation; the original volume is available at the ^[ref].

Every group representation of a finite group over a field whose characteristic does not divide the number of elements of the group is semisimple. Given an invariant subspace, averaging any projection onto it over the group produces an equivariant projection, whose kernel is an invariant complement.

The polynomial identities remain true in positive characteristic of a field, but recovering elementary symmetric polynomials by division can fail. In the prime field ${\mathbb{F}}_2$ of characteristic two, the root lists $(0,0)$ and $(1,1)$ have the same power sum zero for every positive exponent: $0^k+0^k=0$ and $1^k+1^k=0$. Their monic polynomials differ, however: $z^2$ and $(z-1{)}^2=z^2+1$. Thus even all positive power sums need not determine the polynomial when its degree meets the characteristic. The relation $2e_2=e_1{p}_1-p_2$ becomes $0=0$ and supplies no value for $e_2$.

Newton's identities let two different summaries of the same polynomial roots talk to each other. Elementary symmetric polynomials record products of distinct root positions; power sums record sums of individual powers, $p_r={\sum }_{i=1}^n{x}_i^r$. Either can be easier to obtain in a particular problem. The identities turn one into the other, often avoiding any need to find the individual polynomial roots.

Write $e_0=1$ and $e_m=0$ for $m>n$. For every positive integer $m$, the identity isIt is a polynomial identity over the integers, so it remains true on substitution in any commutative ring. To solve it for $e_m$ by division, however, $m$ must have a multiplicative inverse. Over a field of characteristic zero, the first $n$ power sums recover every elementary symmetric polynomial. The same recovery works when the positive characteristic of a field is greater than $n$; small characteristic needs care.^[ref]

The first three relations are $e_1=p_1$, $2e_2=e_1{p}_1-p_2$, and $3e_3=e_2{p}_1-e_1{p}_2+p_3$. When division by two and three is permitted, this becomesFor the three roots $1,2,3$, their power sums are $p_1=6$, $p_2=14$, and $p_3=36$. The formulas give $e_2=(36-14)/2=11$ and $e_3=(216-252+72)/6=6$, recovering the polynomial $z^3-6z^2+11z-6$ from aggregate data.

The beauty is that the summaries do not depend on the order of the polynomial roots. In symmetric polynomials, one can work with quantities that survive every relabeling; in linear algebra, traces of matrix powers give the same kind of aggregate information about eigenvalues. The identities convert those summaries into coefficients of a characteristic polynomial.

A short derivation packages the identities into a formal power series. Let $E(t)={\prod }_i(1+x_{i}t)={\sum }_{m\ge 0}{e}_m{t}^m$. Since its constant coefficient is one, it has an inverse as a formal power series, and its formal derivative satisfiesMultiply by $E(t)$ and compare the coefficient of $t^{m-1}$. This gives Newton's identities without a convergence assumption or division by $m$. The division issue arises only when using them to recover $e_m$. The logarithmic-derivative approach is also developed in Newton's Identities (January 2019 notes) by Milan Mossé (§4, pp. 6–8).

The alternate name Newton-Girard identities recalls Albert Girard's earlier work. A modern historical account credits formulas through sixth powers to Albert Girard in 1629 and discusses their later appearance in Isaac Newton's Arithmetica universalis.^[ref] This citation refers to that account; the original historical editions have not been inspected here.

For a nilpotent endomorphism $N$ of a vector space, the commutator action $\mathrm{ad}N:T\mapsto NT-TN$ on its endomorphisms is nilpotent. The commuting left and right multiplication maps give $(\mathrm{ad}N{)}^r(T)={\sum }_{j=0}^r(-1{)}^j\left(\genfrac{}{}{0ex}{}{r}{j}\right)N^{r-j}T{N}^j$. If $N^q=0$ and $r\ge 2q-1$, every summand vanishes. This works in every characteristic of a field and is useful when applying Engel theorem to an Adjoint representation.

A semisimple module is a direct sum of simple modules. A finite-dimensional semisimple algebra is one whose left regular module is semisimple; equivalently, all its finite-dimensional modules are semisimple, or every submodule of a finite-dimensional module has a complementary submodule.

The Maschke theorem states that, for a finite group $G$ and a field $k$ with $|G|\ne 0$ in $k$, every finite-dimensional $kG$-module is semisimple, so $kG$ is a semisimple algebra. The assumption is equivalent to saying that the characteristic of a field does not divide $|G|$. The averaging construction makes a projection onto a submodule equivariant by replacing $P$ with $|G{|}^{-1}{\sum }_{g\in G}gP{g}^{-1}$.

Solutions to all three questions of the University of Oxford Part A linear algebra paper, sat on 20 June 2023. Source: released examination PDF, parent PDF pages 258–260; paper pages 1–3. Question labels follow the paper; the solutions supply their own reasoning. Part 2(c) includes the exceptional case of characteristic of a field three because that part does not specify its field separately.

Let $O=(0:1:0)$ be the point at infinity of the Weierstrass equation of an elliptic curve. It is the identity element for the elliptic curve group law. For an affine rational point $P=(x,y)$, the inverse isIndeed, the two affine intersections with the vertical line have these ordinates, since their sum as roots of the quadratic in $y$ is $-a_{1}x-a_3$. This reflection is valid for a generalized Weierstrass equation of an elliptic curve in every characteristic of a field.

For distinct $P,Q$, draw their line; for $P=Q$, draw the tangent line. Let $R$ be its third intersection with the elliptic curve, counted with intersection multiplicity. Then the sum is the reflection of the third intersection:A vertical line has third intersection $O$, so $P+(-P)=O$. If a tangent has third intersection $O$, then $2P=O$. Define addition with $O$ by $P+O=P$; these conventions include all exceptional cases of the affine construction.

The construction is defined over the coefficient field: after two known intersections, the remaining root is rational over that field. It gives a commutative group operation. To see why it is associative, use the elliptic curve group law from Riemann-Roch: the map $P\mapsto [P-O]$ identifies the elliptic curve with its degree-zero divisor class group. A line has intersection divisor $P+Q+R$ linearly equivalent to $3O$, the divisor of the line at infinity. Thus $[P-O]+[Q-O]+[R-O]=0$. The geometric sum agrees with addition of divisor classes, which is associative.

Work near the identity $O$ of a generalized Weierstrass equation of an elliptic curve. On the projective chart where $Y\ne 0$, use the local parameter $t=-x/y$ and the auxiliary coordinate $w=-1/y$. Substitution givesThe coefficient of $w$ in the linearized equation at $(0,0)$ is one. Consequently, successive coefficient comparison determines a unique formal power series $w(t)$, without dividing by any coefficient of the equation. Its initial terms areRecover the affine coordinates as Laurent series $x=t/w(t)$ and $y=-1/w(t)$; in particular $x=t^{-2}+O(t^{-1})$ and $y=-t^{-3}+O(t^{-2})$.

The elliptic curve group law is a regular morphism near $(O,O)$. Applying the local parameter to the sum and passing to completed local rings producesOne can construct the series without any nonunit division. PutThe divided difference is an integral formal power series, including its tangent line specialization at $T_1=T_2$. The line $w=\lambda t+\nu$ meets the cubic in the two chosen points and a third point $R$. Substitution in the equation shows that its cubic and quadratic coefficients in $t$ are $-A,-B$, whereSince $\lambda$ has total degree at least two and $\nu$ at least three, $A$ is a unit. The sum of the roots therefore givesThe last expression reflects the third intersection according to the chord-and-tangent group law. Both denominators have constant term one. This proves integrality of the universal construction and also provides an explicit coefficient algorithm. Expanding it, the resulting formal group of an elliptic curve beginsThe integral coefficient recursion and the unit-denominator addition calculation work universally over $\mathbb{Z}[a_1,a_2,a_3,a_4,a_6]$; thus these coefficients specialize in every characteristic of a field. This is the initial coefficients of the elliptic formal group at infinity construction, rather than a definition requiring division by the elliptic-curve discriminant.

The identity, commutativity and associativity of the elliptic curve group law give the defining identities of a one-dimensional commutative formal group law:Reflection in the generalized Weierstrass equation of an elliptic curve gives its formal inverse explicitly:Successive addition gives the multiplication series $[n{]}_F(T)=nT+O(T^2)$.

In characteristic zero, an especially useful description comes from the invariant differential on an elliptic curve:Integrating with zero constant term constructs the formal logarithmTranslation invariance of the invariant differential on an elliptic curve implies $L(F(X,Y))=L(X)+L(Y)$. Thus the formal logarithm converts the formal group law into the formal additive group in characteristic zero. Expanding this identity to degree three also recovers the displayed coefficients of $F$; in positive characteristic the integral addition construction remains valid although the logarithmic denominators may not exist.