# Separable extension

A **separable extension** is an algebraic field extension \( E/F \) in which every element of \( E \) has a minimal polynomial over \( F \) with no repeated roots. An element \( \alpha \) is separable over \( F \) exactly when its minimal polynomial over \( F \) is separable, and the extension is separable when all of its elements are<sup>[1](https://www.math.ucla.edu/~sharifi/notes/algebra-ch06.html)</sup>. Separability is one of the two conditions, together with normality, that make a finite algebraic extension a [Galois extension](https://www.edgepedia.org/galois-extension), and it is the condition that controls how many embeddings an extension admits<sup>[1](https://www.math.ucla.edu/~sharifi/notes/algebra-ch06.html)</sup>.

| Key fact | Detail |
|---|---|
| Element | \( \alpha \) is separable over \( F \) iff its minimal polynomial over \( F \) is separable (no multiple roots)<sup>[1](https://www.math.ucla.edu/~sharifi/notes/algebra-ch06.html)</sup> |
| Extension | An algebraic extension \( E/F \) is separable iff every \( \alpha \in E \) is separable over \( F \)<sup>[1](https://www.math.ucla.edu/~sharifi/notes/algebra-ch06.html)</sup> |
| Derivative test | An irreducible \( f \in F[x] \) is inseparable iff \( f' = 0 \), which forces characteristic \( p > 0 \) and \( f(x) = g(x^p) \)<sup>[2](https://stacks.math.columbia.edu/tag/09GZ)</sup> |
| Embedding count | For finite \( E/F \), the number of embeddings into \( \overline{F} \) fixing \( F \) divides \( [E:F] \), with equality iff \( E/F \) is separable<sup>[1](https://math.ucla.edu/~sharifi/notes/algebra-ch06.html)</sup> |
| Perfect fields | A field is perfect iff it has characteristic 0, or characteristic \( p > 0 \) with every element a \( p \)th root; all its algebraic extensions are separable<sup>[3](https://stacks.math.columbia.edu/tag/05DU)</sup> |
| Canonical example | \( x^p - t \in \mathbb{F}_p(t)[x] \) is irreducible but equals \( (x - t^{1/p})^p \), giving an inseparable element<sup>[1](https://www.math.ucla.edu/~sharifi/notes/algebra-ch06.html)</sup> |
| Primitive element | Every finite separable extension is simple, \( E = F(\alpha) \)<sup>[1](https://math.ucla.edu/~sharifi/notes/algebra-ch06.html)</sup> |
| Origin | Separable elements and perfect fields were introduced by Ernst Steinitz in *Algebraische Theorie der Körper* (1910)<sup>[4](https://mathshistory.st-andrews.ac.uk/DSB/Steinitz.pdf)</sup> |

## Polynomials and elements

The definitions build upward from polynomials. A nonconstant polynomial \( f \in F[x] \) is separable if every root of \( f \) in an algebraic closure \( \overline{F} \) has multiplicity 1, where the multiplicity of \( \alpha \) is the largest \( m \) such that \( (x - \alpha)^m \) divides \( f \)<sup>[1](https://www.math.ucla.edu/~sharifi/notes/algebra-ch06.html)</sup>. An element \( \alpha \in \overline{F} \) is then separable over \( F \) if and only if its minimal polynomial over \( F \) is separable<sup>[1](https://www.math.ucla.edu/~sharifi/notes/algebra-ch06.html)</sup>. For an irreducible polynomial, all roots in \( \overline{F} \) have the same multiplicity, so the number of distinct roots divides the degree<sup>[1](https://www.math.ucla.edu/~sharifi/notes/algebra-ch06.html)</sup>.

The derivative gives the practical test. For an irreducible \( f \in F[x] \), exactly one of two things happens: \( f \) is relatively prime to its derivative \( f' \), or \( f' \) is the zero polynomial<sup>[2](https://stacks.math.columbia.edu/tag/09GZ)</sup>. The second case can only occur in characteristic \( p > 0 \), where it forces \( f(x) = Q(x^q) \) for a power \( q = p^f \) of \( p \) and an irreducible \( Q \) relatively prime to its derivative<sup>[2](https://stacks.math.columbia.edu/tag/09GZ)</sup>. In characteristic 0 this can never happen, so every irreducible polynomial over a field of characteristic 0 is separable, and every algebraic extension of such a field is separable<sup>[2](https://stacks.math.columbia.edu/tag/09GZ)</sup>.

## The characteristic p phenomenon

Inseparability exists only in positive characteristic. The standard example is \( f = x^p - t \in \mathbb{F}_p(t)[x] \), which is irreducible, yet over the algebraic closure it factors as \( f = (x - t^{1/p})^p \), so \( t^{1/p} \) has multiplicity \( p \) as a root<sup>[1](https://www.math.ucla.edu/~sharifi/notes/algebra-ch06.html)</sup>. More generally, if \( \alpha \) is inseparable over \( k \) with minimal polynomial \( f \), then \( f(X) = g(X^{p^n}) \) for some \( n \geq 1 \) and irreducible \( g \); choosing \( n \) maximal makes \( g \) separable and makes \( \alpha^{p^n} \) separable over \( k \)<sup>[5](https://www.cis.upenn.edu/~cis6100/alg4.pdf)</sup>.

The opposite extreme is a **purely inseparable extension**: in characteristic \( p > 0 \), an element \( \alpha \) is purely inseparable over \( F \) if \( \alpha^q \in F \) for some power \( q \) of \( p \), and the extension is purely inseparable when every element is<sup>[6](https://stacks.math.columbia.edu/tag/09HD)</sup>. Such extensions only appear in positive characteristic<sup>[6](https://stacks.math.columbia.edu/tag/09HD)</sup>. For a finite extension \( K/F \), the separable elements form an intermediate subfield \( E \) that is maximal for separability, and \( K/E \) is purely inseparable; the degrees \( [E:F] \) and \( [K:E] \) are called the separable degree \( [K:F]_s \) and inseparable degree \( [K:F]_i \)<sup>[1](https://math.ucla.edu/~sharifi/notes/algebra-ch06.html)</sup>. In characteristic \( p \), the inseparable degree is a power of \( p \)<sup>[1](https://math.ucla.edu/~sharifi/notes/algebra-ch06.html)</sup>.

## Perfect fields

A field \( k \) is **perfect** if every field extension of \( k \) is separable over \( k \)<sup>[3](https://stacks.math.columbia.edu/tag/05DU)</sup>. This happens exactly when \( k \) has characteristic 0, or characteristic \( p > 0 \) in which every element has a \( p \)th root, so that \( k = k^{1/p} \)<sup>[3](https://stacks.math.columbia.edu/tag/05DU)</sup>. Finite fields are perfect: \( \mathbb{F}_{p^n} \) consists of the \( p^n \) distinct roots of \( x^{p^n} - x \), so the minimal polynomial of any element divides a polynomial with distinct roots<sup>[1](https://math.ucla.edu/~sharifi/notes/algebra-ch06.html)</sup>. By contrast, \( k(T) \) with \( \operatorname{char}(k) = p > 0 \) is always imperfect<sup>[5](https://www.cis.upenn.edu/~cis6100/alg4.pdf)</sup>.

Every field can be made perfect by force. For each \( n \) there is a unique algebraic extension \( k^{1/p^n} \) in which every element of \( k \) has a \( p^n \)th root, and the union \( k^{\mathrm{perf}} \) of these is a purely inseparable extension of \( k \) that is perfect, unique up to unique isomorphism; it is called the perfect closure<sup>[3](https://stacks.math.columbia.edu/tag/05DU)</sup>.

## Embeddings and the primitive element theorem

Separability has a concrete meaning in terms of field homomorphisms. For a finite extension \( E/F \) with algebraic closure \( \overline{F} \), the number of embeddings of \( E \) into \( \overline{F} \) fixing \( F \) divides \( [E:F] \), and it equals \( [E:F] \) if and only if \( E/F \) is separable<sup>[1](https://math.ucla.edu/~sharifi/notes/algebra-ch06.html)</sup>. The separable degree \( [K:F]_s \) of a finite extension equals exactly this number of embeddings<sup>[1](https://math.ucla.edu/~sharifi/notes/algebra-ch06.html)</sup>.

Separability also behaves well under composition. If \( K/E/F \) is a tower of algebraic extensions, then \( K/F \) is separable if and only if both \( K/E \) and \( E/F \) are<sup>[1](https://math.ucla.edu/~sharifi/notes/algebra-ch06.html)</sup>. Consequently, to check separability of a finitely generated extension it suffices to check it on a generating set<sup>[1](https://math.ucla.edu/~sharifi/notes/algebra-ch06.html)</sup>.

The main structural payoff is the **primitive element theorem**: every finite separable extension is simple, meaning \( E = F(\alpha) \) for a single element \( \alpha \), called a primitive element<sup>[1](https://math.ucla.edu/~sharifi/notes/algebra-ch06.html)</sup>. More generally, a finite extension \( E/F \) has a primitive element if and only if there are finitely many intermediate fields \( E/K/F \), and both conditions hold when \( E/F \) is separable<sup>[7](https://stacks.math.columbia.edu/tag/09HZ)</sup>.

## Separable closure and Galois theory

Inside a fixed algebraic closure \( \overline{k} \), the set of elements separable over \( k \) forms the **separable closure** \( k^{\mathrm{sep}} \)<sup>[8](https://web.math.princeton.edu/~shouwu/publications/LAG2.pdf)</sup>. Every algebraic extension \( L/k \) decomposes uniquely as a separable extension followed by a purely inseparable one, with \( k^{\mathrm{sep}} \) as the intermediate field<sup>[8](https://web.math.princeton.edu/~shouwu/publications/LAG2.pdf)</sup><sup> • </sup><sup>[9](https://stacks.math.columbia.edu/tag/030K)</sup>.

Separability is what makes the automorphism group of an extension large enough to support [Galois theory](https://www.edgepedia.org/galois-theory). A finite extension is Galois when it is both normal and separable, and for a separable extension of degree \( n \) generated by one element, the discriminant of the minimal polynomial is nonzero, being a product of squared differences \( \prod_{i<j}(\alpha_j - \alpha_i)^2 \) of the \( n \) distinct conjugates<sup>[10](https://www.math.ucla.edu/~sharifi/notes/algebra-ch10.html)</sup>.

## Wider algebraic context

The notion extends beyond fields. Étale ring maps of the form \( k \to S \) are exactly those for which \( S \), viewed as a \( k \)-algebra, is isomorphic to a product of finitely many finite separable field extensions of \( k \); the proof uses that a finite separable extension \( k'/k \) with \( k' = k(\alpha) \cong k[x]/(f) \) has \( \gcd(f, f') = 1 \), which is what makes the map étale<sup>[11](https://stacks.math.columbia.edu/tag/00U3)</sup>. For nonalgebraic extensions, a field \( K \) is called separably generated over \( k \) if it has a transcendence basis \( \{x_i\} \) such that \( K/k(x_i) \) is separable algebraic, and a separably generated extension is separable in the general sense<sup>[12](https://stacks.math.columbia.edu/tag/030I)</sup>.

## History

The concepts of separable element and perfect field were introduced by the German mathematician [Ernst Steinitz](https://www.edgepedia.org/ernst-steinitz) in his 1910 paper *Algebraische Theorie der Körper* (Algebraic Theory of Fields), published in Crelle's Journal, which also gave the first proof of the existence and uniqueness of the algebraic closure of a field and is regarded as the origin of the modern abstract view of algebra<sup>[4](https://mathshistory.st-andrews.ac.uk/DSB/Steinitz.pdf)</sup><sup> • </sup><sup>[13](https://www.deutsche-biographie.de/gnd117257850.html)</sup>.

## References

1. [Romyar Sharifi, Abstract Algebra, Chapter 6: Field theory and Galois theory, UCLA lecture notes](https://www.math.ucla.edu/~sharifi/notes/algebra-ch06.html)
2. [The Stacks Project, Section 9.12 (09GZ): Separable algebraic extensions](https://stacks.math.columbia.edu/tag/09GZ)
3. [The Stacks Project, Section 10.45 (05DU): Perfect fields](https://stacks.math.columbia.edu/tag/05DU)
4. [Ernst Steinitz, Complete Dictionary of Scientific Biography](https://mathshistory.st-andrews.ac.uk/DSB/Steinitz.pdf)
5. [University of Pennsylvania, Algebra course notes, Chapter 4: Fields and Galois theory](https://www.cis.upenn.edu/~cis6100/alg4.pdf)
6. [The Stacks Project, Section 9.14 (09HD): Purely inseparable extensions](https://stacks.math.columbia.edu/tag/09HD)
7. [The Stacks Project, Section 9.19 (09HZ): Primitive elements](https://stacks.math.columbia.edu/tag/09HZ)
8. [Shou-Wu Zhang, Preliminary field theory, Princeton lecture notes](https://web.math.princeton.edu/~shouwu/publications/LAG2.pdf)
9. [The Stacks Project, Lemma 9.14.6 (030K)](https://stacks.math.columbia.edu/tag/030K)
10. [Romyar Sharifi, Abstract Algebra, Chapter 10: Topics in Galois theory, UCLA lecture notes](https://www.math.ucla.edu/~sharifi/notes/algebra-ch10.html)
11. [The Stacks Project, Lemma 10.143.4 (00U3)](https://stacks.math.columbia.edu/tag/00U3)
12. [The Stacks Project, Section 10.42 (030I): Separable extensions](https://stacks.math.columbia.edu/tag/030I)
13. [Deutsche Biographie: Steinitz, Ernst](https://www.deutsche-biographie.de/gnd117257850.html)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Algebraic structures › Field and Galois theory › Fields and field extensions*

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