Field

A field is a commutative ring with $0\ne 1$ in which every nonzero element has a multiplicative inverse. Examples include the rational numbers, real numbers, and complex numbers. The integers are a ring but not a field: dividing $1$ by $2$ leaves them. For a prime number $p$, the prime field ${\mathbb{F}}_p=\mathbb{Z}/p\mathbb{Z}$ is a finite example, because every nonzero residue is coprime to $p$ and hence has a modular inverse.

In ${\mathbb{F}}_5$, the nonzero inverse pairs are $1^{-1}=1$, $2^{-1}=3$, $3^{-1}=2$, and $4^{-1}=4$. Thus $2x=3$ has solution $x=3\cdot 3=4$ in this field; the real number solution would be $3/2$. Dividing by a nonzero coefficient is valid in both settings, but it means division according to the chosen field's operations.

A field has no zero divisors: if $ab=0$ and $a\ne 0$, multiplying by $a^{-1}$ gives $b=0$. This short argument supports polynomial factorization arguments and pivot division in Gaussian elimination. It does not hold in every commutative ring: modulo $6$, the nonzero residues $2$ and $3$ multiply to zero. The distinction is visible in the figure under algebraic structure.

A field is $C_i$ if every homogeneous polynomial of degree $d\ge 1$ in more than $d^i$ variables has a nonzero zero. Algebraically closed fields are $C_0$, finite fields are $C_1$, and the Lang-Nagata theorem for Ci fields controls function-field extensions.

A finitely generated field extension of transcendence degree $s$ of a $C_i$ field is $C_{i+s}$. In particular a function field of a curve over a C1 field is $C_2$. The theorem includes stability under finite algebraic extensions when $s=0$.

A $C_2$ field is a Ci field with the bound $N>d^2$. Function fields of curves over finite fields are examples. The extra-variable construction in surjectivity of reduced norms over C2 fields makes this polynomial-solvability property useful for division algebras.

For a central division algebra $D$ of degree $d$ over a C2 field, the form $\mathrm{Nrd}(X)-a{T}^d$ has degree $d$ in $d^2+1$ variables. Its nontrivial zero must have $T\ne 0$, because the reduced norm of a division algebra is anisotropic. Dividing $X$ by $T$ gives norm $a$. Zero is the norm of zero.

A $C_1$ field is a Ci field with the bound $N>d$. The Chevalley-Warning theorem gives finite-field examples; the Tsen theorem gives function fields of curves over algebraically closed fields. A $C_1$ field has trivial Brauer group.

A central division algebra of degree $d>1$ has an anisotropic reduced norm of degree $d$ in $d^2>d$ variables. That contradicts the C1 field property. Thus only degree one is possible, and every central simple algebra is split.

For a degree-$m$ extension $L/K$, substitute a $K$-basis of $L$ into a homogeneous degree-$d$ polynomial in $N>d$ variables over $L$, then take its field norm. The resulting degree-$md$ polynomial has $mN>md$ variables over $K$. A nonzero zero over a C1 field gives a nonzero zero of the original form because the norm of a field element vanishes only at zero. Separability is unnecessary.

The function field of a curve over an algebraically closed field is a C1 field. Consequently its Brauer group is zero. This is the input for cyclic splitting by extension of constants over a finite constant field.

For the function field $K$ of a curve over a finite full constant field $k$, extend constants to $\overline{k}$. The resulting curve function field is $C_1$ by the Tsen theorem, so every central simple algebra splits there. The finite collection of coefficients of a splitting isomorphism descends to some finite constant extension $k^{\prime }/k$. Regularity makes $K{k}^{\prime }/K$ cyclic, giving a finite cyclic splitting field.

A field is formally real when $-1$ is not a finite sum of squares. The Artin-Schreier ordering criterion says this is equivalent to the existence of an ordering compatible with the field operations. Such a field has characteristic zero and embeds into a real closed field.

A field can be ordered compatibly with addition and multiplication if and only if it is a formally real field. This ordering criterion concerns real algebra, and is different from Artin–Schreier theory of cyclic extensions in positive characteristic.

A real closed field is a formally real field with no proper formally real algebraic extension. Equivalently every positive element in its unique compatible order is a square and every odd-degree polynomial has a root. Its positive cone is definable in the field language as the nonzero squares. The Theory of real closed fields is model-complete.

A real closure of an ordered field $K$ is a real closed algebraic extension whose order extends the chosen order of $K$. Existence is the real closure theorem; uniqueness holds up to ordered field isomorphism over $K$. A different ordering of a formally real field can produce a different ordered real closure.

Every ordered field has an ordered algebraic real closure. Combined with the Artin-Schreier ordering criterion, every formally real field embeds into a real closed field. This is one embedding direction in the model companion relation between their theories.

A valuation assigns sizes to elements of a field in a way compatible with multiplication and addition. In additive notation, $v(xy)=v(x)+v(y)$ and $v(x+y)\ge \min \{v(x),v(y)\}$.

Two additive valuations are equivalent when they have the same valuation ring. Their ordered value-group images then identify compatibly with the valuations; for nontrivial real-valued rank-one valuations this is multiplication by a positive constant.

A nontrivial additive valuation has rank one when its ordered value group is Archimedean and therefore embeds order-preservingly in $\mathbb{R}$. Exponentiating a real-valued representative gives a Non-Archimedean absolute value; changing the embedding rescales the valuation positively. Higher-rank ordered value groups cannot be represented by real-valued absolute values on a field.

For a polynomial over a discretely valued field and a rational weight $t$, this coefficient-minimum function is multiplicative. Scale the variable by an element of valuation $t$ in a valued extension and reduce the initial terms; nonzero residue polynomials have nonzero product. For a monic polynomial factored into roots, $w_t(f)={\sum }_r\min (t,v(r))$. Its supporting lines determine the lower Newton polygon, giving the Newton polygon root valuation theorem and its multiplicities.

A non-Archimedean valuation satisfies the strong triangle inequalityIn multiplicative absolute-value notation this becomes $|x+y|\le \max \{|x|,|y|\}$.

An additive nontrivial valuation on the rational field is equivalent to exactly one P-adic valuation. Integers have nonnegative values; some prime must have positive value, and Bézout's identity prevents two distinct primes from doing so. Unique factorization then gives the formula, with $c>0$ in the real-valued case. The trivial valuation is a separate possibility if admitted by the definition.

A field is algebraically closed when every nonconstant polynomial over it has a root in it, equivalently when every such polynomial splits into linear factors.

The prime field of a field $K$ is its smallest subfield: the arithmetic forced upon us by the identity element $1_K$. Every subfield must contain $1_K$, its repeated sums, their additive inverses, and the multiplicative inverses of nonzero results. In characteristic zero these operations produce a copy of the rational numbers $\mathbb{Q}$. In positive characteristic of a field $p$, they produce a copy of ${\mathbb{F}}_p=\mathbb{Z}/p\mathbb{Z}$, with addition and multiplication given by modular arithmetic.

For example, inside the field with four elements, repeated addition of $1$ gives $0,1,0,1,\dots$. Thus its prime field is $\{0,1\}$, and the other two elements cannot be reached by repeatedly adding $1$. This is why a finite field need not consist entirely of ordinary integer residues: a larger finite field extends the arithmetic of its prime field with additional elements.

The positive characteristic of a field must be a prime number. If its least vanishing multiple of $1_K$ were $rs$ with $1<r,s<rs$, then $(r{1}_K)(s{1}_K)=0$ would exhibit zero divisors. The absence of zero divisors rules this out. The prime field supplies the scalars when viewing $K$ as a vector space; its basis then explains the possible sizes of a finite field.

A subfield of a field $K$ is a subset containing the additive and multiplicative identities and closed under addition, subtraction, multiplication, and inversion of nonzero elements. It is itself a field under the operations inherited from $K$.

The characteristic of a field $F$ is the least positive integer $n$ for which $n\cdot 1_F=0$, or zero if no such integer exists.

In a field of characteristic two, $1+1=0$. The usual splitting of a tensor product into its symmetric square and exterior square by $(I\pm \tau )/2$ is unavailable.

A field has characteristic zero when no positive integer multiple of its identity vanishes. Equivalently, its prime field is isomorphic to $\mathbb{Q}$. Every irreducible polynomial over such a field is a separable polynomial.

A field is perfect when every algebraic extension is separable. In positive characteristic $p$, this is equivalent to surjectivity of the Frobenius endomorphism $x\mapsto x^p$; every finite field is perfect.