Existence of Separable Maximal Subfields

Q: Why does the proof need the centralizer at all — why not apply Noether–Jacobson directly to D over F?

Applied to D over F it produces one separable element, giving a separable subfield of some degree, with no control on size. To grow that subfield one must know what commutes with it, and the double centralizer identity Z(C_D(K)) = K is what makes the centralizer a new division algebra with the old subfield as its exact centre, so the same theorem can be applied again one level down.

Q: Does (15.12) hold for algebraic division algebras that are not centrally finite?

It is not known in that generality — Lam states the question explicitly and reports it open. The obstruction is structural: the proof consumes the Double Centralizer Theorem, which needs _F K < ∈fty, and there is no known substitute.

Q: Can a central division algebra have *only* purely inseparable maximal subfields?

No, if it is centrally finite — that is exactly what (15.12) forbids. It can certainly have some, as the degree-p symbol algebra shows, but a separable one always coexists with them.

Q: How does this relate to the crossed product problem?

Separable splitting fields make Galois cohomology available and show every Brauer class is represented by a crossed product. Whether the division algebra D itself contains a Galois maximal subfield is a strictly stronger question, answered negatively in general by Amitsur in 1972.

Q: Where is transitivity of separability used, and could it be avoided?

At the final step, to see that K(z) — separable over K, which is separable over F — is separable over F. It cannot be avoided: the recursion produces improvements relative to K, and the maximality hypothesis is relative to F.

Q: Is there an analogue for maximal subfields that are Galois, or cyclic?

No. Both fail: Amitsur's examples have no Galois maximal subfield, and even among crossed products not all are cyclic. The proof method breaks at the first move — being Galois over F is not preserved when a separable element is adjoined inside a centralizer.

Executive Summary

Let $D$ be a division ring finite-dimensional over its centre $F$ . Then $D$ contains a maximal subfield that is separable over $F$ , and more precisely every subfield of $D$ separable over $F$ can be enlarged to such a maximal subfield. In characteristic zero the statement is empty — every algebraic extension is separable — so its content is entirely in characteristic $p$ .

The proof is a two-line contradiction resting on two heavy imports: the Double Centralizer Theorem, which identifies $Z (C_{D} (K))$ with $K$ , and the Noether–Jacobson theorem, which produces a separable element in any noncommutative algebraic division algebra. If a maximal separable subfield $K$ failed to be a maximal subfield, its centralizer would be a noncommutative division algebra with centre exactly $K$ , and Noether–Jacobson would hand back a separable element outside $K$ .

$r$ Degree of the separable maximal subfield

char $p$ Where the theorem has content

$\dim_{F} D < \infty$ Hypothesis that cannot be dropped

Crossed productsMain downstream consequence

Overview

Maximal subfields exist in profusion — Zorn's Lemma sees to that — but existence alone is not usable. Almost every structural description of a central division algebra requires a maximal subfield with extra properties: separable, so that the primitive element theorem applies; Galois, so that a crossed product presentation exists; cyclic, so that a symbol presentation exists. Each level of demand is strictly harder to satisfy, and the first is exactly what $(15.12)$ delivers.

E \subseteq D separable over F ⟹ \exists K maximal subfield of D, E \subseteq K, K / F separable .

(15.12)

The enlargement form of the theorem; taking $E = F$ gives the bare existence statement.

The demand is not vacuous. In characteristic $p$ a central division algebra can perfectly well contain a purely inseparable maximal subfield — the worked example below has one of each — so "pick any maximal subfield" is not a legitimate move in any argument that needs separability.

The hypothesis of central finiteness enters through the Double Centralizer Theorem, which needs $\dim_{F} K < \infty$ . Lam records explicitly that it is unclear how to extend the argument to algebraic division algebras, and states that he does not know whether an algebraic division algebra with centre $F$ must contain a maximal subfield separable over $F$ .

Learning Objectives

State $(15.12)$ in both its forms and identify the standing hypothesis.
Show that a maximal separable subfield exists, using finite dimensionality.
Prove $Z (C_{D} (K)) = K$ from the double centralizer identity.
Complete the contradiction using the Noether–Jacobson theorem.
Verify transitivity of separability at the point where the proof uses it.
Construct a degree- $p$ division algebra with both separable and purely inseparable maximal subfields.

Definitions

Standing hypothesis: $D$ is a division ring with $F = Z (D)$ and $\dim_{F} D = r^{2} < \infty$ ; every subfield mentioned contains $F$ .
Separable over $F$: A subfield $E \subseteq D$ with $E / F$ a separable field extension. Every element of $E$ then has a minimal polynomial over $F$ with distinct roots in a splitting field.
Purely inseparable over $F$: In characteristic $p$ : every $a \in E$ satisfies $a^{p^{n}} \in F$ for some $n$ . Such an extension has degree a power of $p$ and admits no nontrivial $F$ -automorphisms.
$C_{D} (K)$: The centralizer of $K$ in $D$ . When $\dim_{F} K < \infty$ it satisfies $C_{D} (C_{D} (K)) = K$ and $\dim_{F} K \cdot \dim_{F} C_{D} (K) = \dim_{F} D$ .
Maximal separable subfield: A subfield separable over $F$ , maximal among such. It exists by finite dimensionality, and $(15.12)$ says it is in fact a maximal subfield.

Separability is transitive: if $M / K$ and $K / F$ are separable algebraic extensions then so is $M / F$ . This is the step that lets a local improvement inside a centralizer be exported back to the base field.

Core Concepts

The centre of a centralizer

Let $K$ be a subfield of $D$ containing $F$ , with $\dim_{F} K < \infty$ , and let $L = C_{D} (K)$ . Directly from the definition of a centre,

Z (L) = L \cap C_{D} (L) = L \cap K = K,

(15.12a)

The middle equality is the Double Centralizer Theorem $C_{D} (L) = K$ ; the last uses $K \subseteq L$ , which holds because $K$ is commutative.

This identity is the engine of the proof. It says that a non-maximal subfield $K$ sits inside a division algebra $L$ as its exact centre, so $L$ is a division algebra centrally finite over $K$ — and it is noncommutative precisely when $K$ fails to be a maximal subfield of $D$ .

Recursion into the centralizer

Once $L$ is recognised as a noncommutative division algebra with centre $K$ , the Noether–Jacobson theorem applies to $L$ over $K$ : there is an element $z \in L ∖ K$ separable over $K$ . Separability is transitive, so $K (z)$ is separable over $F$ as well — and it is a subfield of $D$ strictly larger than $K$ . That is the contradiction, provided $K$ was chosen maximal among separable subfields.

K

not maximal in

D

L = C_{D} (K) ⊋ K

Z (L) = K

L

noncommutative

\exists z \in L ∖ K

separable over

K

K (z) / F

separable

Why characteristic $p$ is the only interesting case

If $char F = 0$ , or more generally if $F$ is perfect — for instance any finite field, though Wedderburn's little theorem removes that case entirely — every algebraic extension of $F$ is separable, so every maximal subfield already qualifies. In characteristic $p$ with $F$ imperfect, maximal subfields can be purely inseparable, and $(15.12)$ is the guarantee that they are not all of that kind.

Key Results

Theorem(15.12)Existence of separable maximal subfields

Let $D$ be a centrally finite division ring with centre $F$ . Then:

$D$ has a maximal subfield $K$ which is separable over $F$ ;
more precisely, every subfield $E \subseteq D$ that is separable over $F$ is contained in a maximal subfield $K \subseteq D$ separable over $F$ .

Such a $K$ has $\dim_{F} K = r$ where $\dim_{F} D = r^{2}$ , by the degree criterion of $(15.8)$ .

Proof

Statement (1) is the case $E = F$ of statement (2), so we prove (2).

**Choice of $K$ .** Among the subfields of $D$ that contain $E$ and are separable over $F$ , choose one, $K$ , of largest $F$ -dimension. This is possible because all such subfields are $F$ -subspaces of the finite-dimensional space $D$ , so their dimensions form a bounded set of positive integers.

**Claim: $K$ is a maximal subfield of $D$ .** Suppose not. Put $L : = C_{D} (K)$ . By $(15.7)$ , failure of maximality means $L ⊋ K$ . Since $\dim_{F} K \leq \dim_{F} D < \infty$ , the Double Centralizer Theorem $(15.4)$ applies and gives $C_{D} (L) = K$ . Hence

Z (L) = L \cap C_{D} (L) = L \cap K = K .

In particular $L$ is a division ring whose centre is $K$ , and $L \neq K = Z (L)$ , so $L$ is noncommutative. Moreover $L$ is finite-dimensional over $K$ , hence certainly algebraic over $K$ .

Apply Noether–Jacobson. By $(15.11)$ applied to the noncommutative division ring $L$ , algebraic over the field $K \subseteq Z (L)$ , there exists $z \in L ∖ K$ separable over $K$ . Then $K (z)$ is a separable extension of $K$ , and $K$ is separable over $F$ by choice; by transitivity of separability, $K (z)$ is separable over $F$ .

But $K (z)$ is a subfield of $D$ containing $E$ , separable over $F$ , with $K (z) ⊋ K$ and hence of strictly larger $F$ -dimension. This contradicts the choice of $K$ . Therefore $K$ is a maximal subfield of $D$ , separable over $F$ and containing $E$ .

Corollary—A separable primitive element

Let $D$ be centrally finite of degree $r$ over $F$ . Then there is $α \in D$ with $F (α)$ a maximal subfield, separable of degree $r$ over $F$ ; consequently $α$ has a separable minimal polynomial over $F$ of degree exactly $r$ .

Proof. Take $K$ as in $(15.12)$ ; $K / F$ is finite separable, so the primitive element theorem gives $K = F (α)$ , and $\dim_{F} K = r$ by $(15.8)$ . This corollary is the input to the Brauer–Albert basis theorem $(15.16)$ .

Corollary—Splitting by a Galois extension

Let $D$ be centrally finite with centre $F$ . Then $D$ is split by a finite separable extension of $F$ , and hence by a finite Galois extension: if $K$ is a separable maximal subfield then $D \otimes_{F} K ≅ M_{r} (K)$ by $(15.8)$ , and any field containing a splitting field is again a splitting field, so the Galois closure of $K / F$ splits $D$ too.

Consequently the Brauer group of $F$ is the union of the relative Brauer groups $Br (M / F)$ over finite Galois extensions $M / F$ , and every Brauer class is represented by a crossed product algebra. This does not say that $D$ itself is a crossed product — Amitsur constructed central division algebras with no Galois maximal subfield.

Remark—The limits of the argument

Every step past the choice of $K$ consumes finite dimensionality: the Double Centralizer Theorem requires $\dim_{F} K < \infty$ , and the existence of a maximal separable subfield requires bounded dimensions. For an algebraic — but centrally infinite — division algebra the argument gives nothing, and Lam records the existence of a separable maximal subfield in that generality as an open question.

Proof Techniques and Method

The reusable moves behind this proof.

Move 1

Maximise the property, then prove maximality

Do not try to build the object directly. Choose something maximal with respect to the desired property — separability — and then show that maximality for the property forces maximality outright.

Move 2

Descend into the centralizer

The centralizer of a subfield is again a division algebra, and the double centralizer identity makes the subfield its exact centre. This converts one problem about $(K, D)$ into the same problem about $(Z (L), L)$ , one level down.

Move 3

Export by transitivity

An improvement obtained over $K$ is useless unless it is an improvement over $F$ . Transitivity of separability is the bridge, and it is the only property of separability the proof actually needs.

The same three-step shape recurs whenever one wants a maximal subfield with a prescribed property $P$ : check that $P$ is preserved by composita, that the centralizer construction stays inside the category, and that $P$ is transitive. Galois-ness fails the first test, which is exactly why the Galois analogue of $(15.12)$ is false.

Worked Example

A degree- $p$ algebra with maximal subfields of both kinds

Let $k$ be a field of characteristic $p > 0$ and $F = k (s, t)$ the rational function field in two independent variables. Let $K = F (u)$ where $u^{p} - u = s$ : this is an Artin–Schreier extension, cyclic of degree $p$ over $F$ , with Galois group generated by $σ : u \mapsto u + 1$ . Build the cyclic algebra

D = ⨁_{i = 0}^{p - 1} K v^{i}, v^{p} = t, v a v^{- 1} = σ (a) (a \in K), so v u v^{- 1} = u + 1 .

(E.1)

A cyclic algebra of degree $p$ over $F = k (s, t)$ , of dimension $p^{2}$ over its centre.

**Why $D$ is a division ring.** A cyclic algebra $(K / F, σ, t)$ is a division algebra exactly when $t$ is not a norm from $K$ . Give $F$ the $t$ -adic valuation $v_{t}$ with $v_{t} (t) = 1$ . Because $s$ is a $t$ -adic unit, the Artin–Schreier extension $K / F$ is unramified at $v_{t}$ with residue degree $p$ , so every norm from $K$ has $v_{t}$ -value divisible by $p$ . Since $v_{t} (t) = 1$ , $t$ is not a norm, and $D$ is a division algebra of degree $p$ .

Two maximal subfields of the same algebra
Subfield	Defining relation	Degree over $F$	Type	Maximal?
$F (u)$	$u^{p} - u = s$	$p$	separable, cyclic Galois	yes
$F (v)$	$v^{p} = t$	$p$	purely inseparable	yes
$F$	—	$1$	trivially separable	no

Both have degree $p = \sqrt{\dim_{F} D}$ , so both are maximal by the degree criterion of $(15.8)$ . The first is separable — indeed Galois — over $F$ ; the second is purely inseparable, since $x^{p} - t$ is irreducible over $F$ and has $v$ as its only root. So a division algebra can contain maximal subfields of both types simultaneously, and $(15.12)$ is precisely the assurance that the separable type is always available.

The characteristic-zero case is vacuous

For $D = ℍ_{ℚ}$ every maximal subfield is a quadratic field, automatically separable over $ℚ$ . The theorem is true but says nothing new. Its real function is to protect arguments in characteristic $p$ , where the automatic step is not available.

Process and Workflow

Start from what you haveAny subfield

E

separable over

F

— possibly just

F

itself, or a field you need to contain.

Maximise separablyEnlarge

E

within the separable subfields until the

F

-dimension can grow no further. Finite dimensionality guarantees this terminates.

Compute the centralizerForm

L = C_{D} (K)

. If

L = K

K

is already a maximal subfield and you are done.

Recurse if it is notIf

L ⊋ K

, then

Z (L) = K

and

L

is noncommutative; Noether–Jacobson produces a separable element of

L

outside

K

, contradicting maximality. So this branch never occurs.

Read off the degreeThe resulting

K

has

\dim_{F} K = r

and splits

D

D \otimes_{F} K ≅ M_{r} (K)

You need a maximal subfield with a specific property. Which are available?

Separable over $F$ Always available for centrally finite

D

, by

(15.12)

Galois over $F$ Not always. Available iff

D

is a crossed product; Amitsur produced central division algebras where it fails.

Cyclic over $F$ Rarer still. Available for degree

2

and

3

by Wedderburn's theorem on algebras of degree three, and over local and global fields, but not in general.

Purely inseparableOnly in characteristic

p

, and only for special algebras; it is never guaranteed.

Comparison and Classification

How much structure can be demanded of a maximal subfield
	Exists always?	Splits $D$	Gives a presentation of $D$
Some maximal subfield	yes	yes	no
Separable over $F$	yes	yes	partial
Galois over $F$	no	yes	yes
Cyclic over $F$	no	yes	yes
Purely inseparable	no	yes	partial

How much structure can be demanded of a maximal subfield

The two imported theorems
Imported result	Applied to	Supplies	Hypothesis consumed
Double Centralizer $(15.4)$	$K \subseteq D$	$C_{D} (C_{D} (K)) = K$ , hence $Z (L) = K$	$\dim_{F} K < \infty$
Noether–Jacobson $(15.11)$	$L$ over $K$	a separable element of $L ∖ K$	$L$ noncommutative and algebraic over $K$
Maximal subfield theory $(15.7)$ , $(15.8)$	$K \subseteq D$	maximality $\Leftrightarrow$ $C_{D} (K) = K$ ; degree $r$	$\dim_{F} D < \infty$ for the degree statement
Transitivity of separability	$F \subseteq K \subseteq K (z)$	$K (z) / F$ separable	none

Relationship Map

$(15.12)$ sits at a junction: it consumes the whole centralizer theory of the section and feeds the classification results that follow.

Centrally finite $D$ with centre $F$

\dim_{F} D = r^{2}

Maximal subfieldsAll of degree

r

over

F

Separable maximal subfieldsAlways exist, by

(15.12)

Galois maximal subfieldsExist iff

D

is a crossed product — not always

(15.4)

(15.11)

(15.12)

(15.16)

Brauer–AlbertCrossed products and

Br (F)

Applications and Industry Use

Applications here means where this structure is used — inside mathematics and in the engineering and computing disciplines that consume it.

Brauer group

Galois cohomology becomes available

Because every class is split by a finite Galois extension, $Br (F) ≅ H^{2} (Gal (F_{sep} / F), F_{sep}^{\times})$ . Separability of the splitting field is what allows the Galois group — rather than some inseparable substitute — to appear.

Representation theory

Schur index computations

Determining the Schur index of a character means finding a small splitting field. $(15.12)$ guarantees a separable one exists of degree equal to the index, which is what makes the search a finite problem over separable closures.

Computational algebra

Constructing splitting fields

Algorithms that split a central simple algebra look for an element generating a separable maximal subfield, then factor its minimal polynomial. In characteristic $p$ the theorem guarantees the search is not futile.

Coding theory

Cyclic algebra code constructions

Space–time codes require a cyclic maximal subfield, a stronger demand than separability. $(15.12)$ marks the boundary: separability is free, cyclicity must be engineered by choosing the algebra rather than discovered inside it.

Computational Notes

Computational notes cover algorithms, cost and library behaviour rather than manufacturing process.

For $D$ given by structure constants over a field $F$ of characteristic zero, a random element generates a maximal subfield with high probability: its minimal polynomial has degree $r$ generically, and separability is automatic. One randomised element plus one minimal-polynomial computation therefore suffices.
In characteristic $p$ the same random search can repeatedly hit purely inseparable or small-degree elements. $(15.12)$ certifies that separable generators exist, but the proof is non-constructive, so implementations fall back on structured constructions such as symbol algebras.
Once a separable maximal subfield $K = F (α)$ is known, the splitting isomorphism $D \otimes_{F} K ≅ M_{r} (K)$ is computed by writing the left regular representation of $D$ on itself as a right $K$ -space — an $r \times r$ matrix computation over $K$ .
Deciding whether a given central simple algebra has a Galois maximal subfield is far harder and is not a routine library operation; over number fields it is automatic, since every central division algebra over a number field is cyclic.

Failure Modes and Common Mistakes

Do not apply Noether–Jacobson to $D$ over $F$ and expect a separable *subfield of degree $r$ *; it returns a single element, and the recursion into the centralizer is what turns that into a maximal subfield.
Do not forget to check that $L$ is noncommutative before invoking Noether–Jacobson — that hypothesis is exactly what the failure of maximality provides.
Do not assume that a subfield separable over $F$ of degree less than $r$ is contained in a unique separable maximal subfield; the enlargement is far from unique.
Do not read the Galois-splitting corollary as saying that $D$ is a crossed product; only some matrix ring over $D$ carries that structure.

Quick Reference

Statement

D

centrally finite

\Rightarrow

D

has a maximal subfield separable over

F

Stronger formAny separable subfield

E

extends to a separable maximal subfield

Degree

\dim_{F} K = r

where

\dim_{F} D = r^{2}

Key identity

Z (C_{D} (K)) = C_{D} (K) \cap K = K

ToolsDouble Centralizer

(15.4)

and Noether–Jacobson

(15.11)

ContentOnly in characteristic

p

with

F

imperfect

Consequence

D

is split by a finite Galois extension;

Br (F)

is a union of relative Brauer groups

Open in generalUnknown for algebraic, centrally infinite division algebras

Proof skeleton in four lines
Step	Content
1	Choose $K \supseteq E$ separable over $F$ of maximal $F$ -dimension
2	If $K$ is not a maximal subfield, $L = C_{D} (K) ⊋ K$ and $Z (L) = K$
3	$L$ is noncommutative and algebraic over $K$ , so $(15.11)$ gives $z \in L ∖ K$ separable over $K$
4	$K (z) / F$ is separable and larger than $K$ — contradiction, so $K$ is maximal

Frequently Asked Questions

Why does the proof need the centralizer at all — why not apply Noether–Jacobson directly to

D

over

F

Applied to $D$ over $F$ it produces one separable element, giving a separable subfield of some degree, with no control on size. To grow that subfield one must know what commutes with it, and the double centralizer identity $Z (C_{D} (K)) = K$ is what makes the centralizer a new division algebra with the old subfield as its exact centre, so the same theorem can be applied again one level down.

Does

(15.12)

hold for algebraic division algebras that are not centrally finite?

It is not known in that generality — Lam states the question explicitly and reports it open. The obstruction is structural: the proof consumes the Double Centralizer Theorem, which needs $\dim_{F} K < \infty$ , and there is no known substitute.

Can a central division algebra have only purely inseparable maximal subfields?

No, if it is centrally finite — that is exactly what $(15.12)$ forbids. It can certainly have some, as the degree- $p$ symbol algebra shows, but a separable one always coexists with them.

How does this relate to the crossed product problem?

Separable splitting fields make Galois cohomology available and show every Brauer class is represented by a crossed product. Whether the division algebra $D$ itself contains a Galois maximal subfield is a strictly stronger question, answered negatively in general by Amitsur in 1972.

Where is transitivity of separability used, and could it be avoided?

At the final step, to see that $K (z)$ — separable over $K$ , which is separable over $F$ — is separable over $F$ . It cannot be avoided: the recursion produces improvements relative to $K$ , and the maximality hypothesis is relative to $F$ .

Is there an analogue for maximal subfields that are Galois, or cyclic?

No. Both fail: Amitsur's examples have no Galois maximal subfield, and even among crossed products not all are cyclic. The proof method breaks at the first move — being Galois over $F$ is not preserved when a separable element is adjoined inside a centralizer.

References

T. Y. Lam, A First Course in Noncommutative Rings, Graduate Texts in Mathematics 131, Springer-Verlag, 1991, §15, result (15.12) (pp. 257–258).
T. Y. Lam, A First Course in Noncommutative Rings, §15, results (15.4) and (15.11), the two theorems consumed by the proof.
A. A. Albert, Structure of Algebras, American Mathematical Society Colloquium Publications 24, 1939, Chapter VII.
N. Jacobson, Basic Algebra II, 2nd edition, W. H. Freeman, 1989, Chapter 4 (separable splitting fields and crossed products).
P. K. Draxl, Skew Fields, London Mathematical Society Lecture Note Series 81, Cambridge University Press, 1983, Chapters 11–14.
S. A. Amitsur, “On central division algebras”, Israel Journal of Mathematics 12 (1972), 408–420.

AI Suggested Questions

Give the details of the argument that the cyclic algebra with relations $u^{p} - u = s$ , $v^{p} = t$ , $v u v^{- 1} = u + 1$ over $k (s, t)$ is a division algebra.
Prove transitivity of separability for algebraic field extensions.
Show that a maximal subfield of a centrally finite division algebra that is purely inseparable over the centre forces the degree to be a power of the characteristic.
Describe Amitsur's construction of central division algebras that are not crossed products.
How is $Br (F)$ identified with $H^{2}$ of the absolute Galois group, and where does the existence of separable splitting fields enter?
What is known about maximal subfields of algebraic, centrally infinite division algebras?
Design an algorithm that, given structure constants for a central simple algebra in characteristic $p$ , finds a separable maximal subfield.

01Executive Summary

02Overview

03Learning Objectives

04Definitions

05Core Concepts

The centre of a centralizer

Recursion into the centralizer

Why characteristic p is the only interesting case

06Key Results

07Proof Techniques and Method

Maximise the property, then prove maximality

Descend into the centralizer

Export by transitivity

08Worked Example

A degree-p algebra with maximal subfields of both kinds

The characteristic-zero case is vacuous

09Process and Workflow

10Comparison and Classification

11Relationship Map

12Applications and Industry Use

Galois cohomology becomes available

Schur index computations

Constructing splitting fields

Cyclic algebra code constructions

13Computational Notes

14Failure Modes and Common Mistakes

15Quick Reference

16Frequently Asked Questions

17Related KEVOS Topics

18References

19AI Suggested Questions

Continue learning

Executive Summary

Overview

Learning Objectives

Definitions

Core Concepts

Why characteristic $p$ is the only interesting case

Key Results

Proof Techniques and Method

Worked Example

A degree- $p$ algebra with maximal subfields of both kinds

Process and Workflow

Comparison and Classification

Relationship Map

Applications and Industry Use

Computational Notes

Failure Modes and Common Mistakes

Quick Reference

Frequently Asked Questions

Related KEVOS Topics

References

AI Suggested Questions