In
field theory, a branch of
algebra
Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...
, an
algebraic field extension is called a separable extension if for every
, the
minimal polynomial of
over is a
separable polynomial (i.e., its
formal derivative
In mathematics, the formal derivative is an operation on elements of a polynomial ring or a ring of formal power series that mimics the form of the derivative from calculus. Though they appear similar, the algebraic advantage of a formal deriva ...
is not the zero
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...
, or equivalently it has no repeated roots in any extension field).
[Isaacs, p. 281] There is also a more general definition that applies when is not necessarily algebraic over . An extension that is not separable is said to be ''inseparable''.
Every algebraic extension of a
field of
characteristic zero is separable, and every algebraic extension of a
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subt ...
is separable.
[Isaacs, Theorem 18.11, p. 281]
It follows that most extensions that are considered in mathematics are separable. Nevertheless, the concept of separability is important, as the existence of inseparable extensions is the main obstacle for extending many theorems proved in characteristic zero to non-zero characteristic. For example, the
fundamental theorem of Galois theory is a theorem about
normal extension
In abstract algebra, a normal extension is an algebraic field extension ''L''/''K'' for which every irreducible polynomial over ''K'' which has a root in ''L'', splits into linear factors in ''L''. These are one of the conditions for algebraic e ...
s, which remains true in non-zero characteristic only if the extensions are also assumed to be separable.
The opposite concept, a
purely inseparable extension In algebra, a purely inseparable extension of fields is an extension ''k'' ⊆ ''K'' of fields of characteristic ''p'' > 0 such that every element of ''K'' is a root of an equation of the form ''x'q'' = ''a'', wit ...
, also occurs naturally, as every algebraic extension may be decomposed uniquely as a purely inseparable extension of a separable extension. An algebraic extension
of fields of non-zero characteristics is a purely inseparable extension if and only if for every
, the minimal polynomial of
over is ''not'' a separable polynomial, or, equivalently, for every element of , there is a positive
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
such that
.
[Isaacs, p. 298]
The simplest example of a (purely) inseparable extension is
, fields of
rational function
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...
s in the indeterminate ''x'' with coefficients in the
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subt ...
. The element
has minimal polynomial