The Cartan–Brauer–Hua Theorem

Q: Why is there no hypothesis on the characteristic, when the additive analogue (13.7) needs char ≠ 2?

The additive proof adds two expressions to reach 2(a^2c - aca), and that factor of 2 has to be invertible. The multiplicative identity (13.13) produces the element a directly by a division, with no integer coefficient anywhere, so nothing can vanish in characteristic 2.

Q: Does the theorem say anything about normal subgroups of the multiplicative group in general?

Only about those of the form K 0 for a division subring K. Plenty of other normal subgroups exist and are noncentral — the commutator subgroup [D^*,D^*] for a start. The general subgroup structure of D^* is a separate subject, treated on *The Multiplicative Group of a Division Ring*.

Q: Is normality really equivalent to xKx^-1 = K for all x?

Yes. The stated condition xKx^-1 ⊆ K applied to x^-1 gives x^-1Kx ⊆ K, i.e. K ⊆ xKx^-1. Both inclusions give equality, so no generality is lost by stating only the containment.

Q: Why does the proof of (13.10) need Herstein's Lemma at all?

The only obstruction is a finite F(a) sitting in an infinite D. Herstein's Lemma is what converts ‘a is noncentral and torsion in characteristic p’ into a conjugator y with yay^-1 = a^i ≠ a, and a counting argument then forces y to have infinite order — which is where the required infinite field comes from.

Q: Can the theorem be extended to simple rings or to matrix rings?

Not verbatim. Amitsur's example Q(x)[t;] is a simple domain in which the noncentral subfield Q(x) is invariant under all automorphisms. There are valid generalisations to simple and semisimple rings, but each carries extra hypotheses — typically on units or on the invariant subring being a division ring.

Executive Summary

Let $K \subseteq D$ be division rings and suppose $K$ is stable under every inner automorphism of $D$ — equivalently, $K^{*} ⊴ D^{*}$ . The Cartan–Brauer–Hua theorem $(13.17)$ says the only possibilities are $K = D$ and $K \subseteq Z (D)$ . There is no middle ground, and no hypothesis on characteristic, cardinality or dimension is required.

The proof is a single algebraic identity applied twice. Its corollaries are structural: the conjugates of any noncentral element already generate $D$ , and — via the same identity — so do the multiplicative commutators. A companion result on this page, $(13.10)$ , shows that in an infinite division ring no centralizer $C_{D} (a)$ can be finite.

$K \subseteq Z (D)$ Conclusion when

K \neq D

NoneHypothesis on

char D

1947–49Cartan, Brauer, Hua

(13.17)Lam's numbering

Overview

Group theory suggests a naive plan for a Galois theory of skew fields: look for normal division subrings and quotient by them. The Cartan–Brauer–Hua theorem destroys the plan, and that is precisely its value — it tells you the normal division subrings are exactly the ones you already knew about.

K normal in D iff x K x^{- 1} \subseteq K for all x \in D^{*} iff K^{*} ⊴ D^{*}

(13.17a)

Applying the containment to $x^{- 1}$ as well upgrades it to $x K x^{- 1} = K$ for every $x \in D^{*}$ .

Compare the additive analogue proved earlier in the section: a division subring that is a Lie ideal of $D$ is central provided $char K \neq 2$ — see Additive Commutators in Division Rings. The multiplicative statement carries no such caveat, which is one reason it is quoted so often.

The two corollaries below are the working form of the theorem. Both say that a small-looking set of elements is in fact enough to generate everything, and both are proved by the same move: build a division subring out of a conjugation-stable set, observe that it is therefore normal, and let the theorem choose between central and everything.

Learning Objectives

Define normality for a division subring and check it against $D^{*}$ , not merely against a generating set.
Derive the identity $(13.13)$ and see why $b = a - 1$ is the right substitution.
Prove the Cartan–Brauer–Hua theorem $(13.17)$ in full.
Deduce $(13.18)$ : the conjugates of a noncentral $d$ generate $D$ as a division ring.
State $(13.10)$ and explain why every centralizer in an infinite division ring is infinite.
Recognise from Amitsur's example why the theorem cannot be extended verbatim to simple rings.

Definitions

Definition(13.17a)Normal division subring

Let $K \subseteq D$ be a division subring of a division ring $D$ . We call $K$ **normal in $D$ ** if $x K x^{- 1} \subseteq K$ for every $x \in D^{*}$ ; equivalently, $K^{*}$ is a normal subgroup of $D^{*}$ . Since the condition holds for $x^{- 1}$ as well, it forces $x K x^{- 1} = K$ for all $x \in D^{*}$ .

$D^{*}$: The multiplicative group $D ∖ {0}$ of the division ring $D$ .
$Z (D)$: The centre ${z \in D : z d = d z \forall d \in D}$ , a field.
$C_{D} (S)$: The centralizer of a subset $S$ : ${d \in D : d s = s d \forall s \in S}$ . It is a division subring of $D$ containing $Z (D)$ .
$F (S)$: For a subfield $F \subseteq D$ and $S \subseteq D$ , the smallest division subring of $D$ containing $F \cup S$ .
Multiplicative commutator: An element $x^{- 1} y^{- 1} x y$ with $x, y \in D^{*}$ .

Throughout, D is a division ring with identity, K denotes a division subring (so 1 ∈ K), and F = Z(D) unless stated otherwise.

Lemma(13.17b)Commuting sets generate fields

Let $S \subseteq D$ be a set whose elements commute with one another and with a subfield $F \subseteq Z (D)$ . Then $F (S)$ is a field.

Reason. $S$ lies in the division subring $C_{D} (S)$ , so $F (S) \subseteq C_{D} (S)$ ; hence every element of $F (S)$ commutes with every element of $S$ , i.e. $S \subseteq C_{D} (F (S))$ . As $C_{D} (F (S))$ is again a division subring containing $F \cup S$ , it contains $F (S)$ , so $F (S)$ commutes with itself.

Core Concepts

The identity that does all the work

Fix two elements $a, c \in D$ that do not commute. Then $a \neq 0$ , $a \neq 1$ , so $b : = a - 1 \in D^{*}$ , and $b$ fails to commute with $c$ exactly as $a$ does. Substituting $a = b + 1$ gives

a (a^{- 1} c a - b^{- 1} c b) = c a - a b^{- 1} c b = c (b + 1) - (b + 1) b^{- 1} c b = c - b^{- 1} c b \neq 0 .

(13.13)

The right-hand side is nonzero precisely because $b = a - 1$ does not commute with $c$ .

Read the identity as a formula that *solves for $a$ *. Since the left-hand side is nonzero, the bracket is nonzero and invertible, so

a = (c - b^{- 1} c b) (a^{- 1} c a - b^{- 1} c b)^{- 1}, b = a - 1 .

(13.13')

Every ingredient on the right is built from $c$ and its conjugates.

That is the whole point: ** $a$ is expressed rationally in terms of $c$ , $a^{- 1} c a$ and $b^{- 1} c b$ .** If a division subring $K$ happens to contain $c$ and to be stable under conjugation, it contains all three ingredients, hence contains $a$ — however $a$ was chosen.

Why the substitution $b = a - 1$

Two competing demands must be met at once. We need a second element whose conjugation behaviour we control, and we need the difference $c - b^{- 1} c b$ to be nonzero. Adding $1$ to $a$ preserves non-commutation with $c$ (the commutator is unchanged) while creating the algebraic relation $a = b + 1$ that collapses the middle term. No other simple perturbation of $a$ does both.

a \notin K

c \in K

a c \neq c a

b = a - 1 \in D^{*}

identity

(13.13)

a \in K

contradiction

From elementwise commuting to central

The identity only yields *“every $a \notin K$ commutes with every $c \in K$ ”*. Upgrading this to $K \subseteq Z (D)$ costs one further line: given $c \in K$ and $c' \in K^{*}$ , pick any $a \in D ∖ K$ (possible because $K \neq D$ ). Then $a c' \notin K$ as well, so $a$ and $a c'$ both commute with $c$ , and therefore so does $c' = a^{- 1} (a c')$ . Hence $c$ commutes with $K^{*}$ , with $0$ , and with $D ∖ K$ — that is, with all of $D$ .

Key Results

Theorem(13.17)Cartan–Brauer–Hua

Let $D$ be a division ring and let $K \subseteq D$ be a division subring which is normal in $D$ , i.e. $x K x^{- 1} \subseteq K$ for every $x \in D^{*}$ . If $K \neq D$ , then $K \subseteq Z (D)$ .

Equivalently: a division subring invariant under every inner automorphism of $D$ is either all of $D$ or central. No assumption is made on $char D$ , on $\dim_{Z (D)} D$ , or on the cardinality of $D$ .

Proof

**Step 1: every $a \in D ∖ K$ commutes with every $c \in K$ .** Suppose not, so $a c \neq c a$ for some such $a$ and $c$ . Since $1 \in K$ and $a \notin K$ we have $a \neq 1$ and $a \neq 0$ , so $b : = a - 1 \in D^{*}$ ; moreover $b c \neq c b$ . Identity $(13.13)$ gives

a (a^{- 1} c a - b^{- 1} c b) = c - b^{- 1} c b \neq 0,

so the bracket is a nonzero element of $D$ and $a = (c - b^{- 1} c b) (a^{- 1} c a - b^{- 1} c b)^{- 1}$ . Normality gives $a^{- 1} c a \in K$ and $b^{- 1} c b \in K$ , and $c \in K$ ; since $K$ is closed under subtraction, multiplication and inversion of nonzero elements, $a \in K$ . This contradicts $a \notin K$ .

**Step 2: $K$ is central.** Fix $c \in K$ ; we show $c$ commutes with every element of $D$ . Because $K \neq D$ we may choose $a \in D ∖ K$ . For any $c' \in K^{*}$ the product $a c'$ again lies outside $K$ — otherwise $a = (a c') c'^{- 1} \in K$ . By Step 1, both $a$ and $a c'$ commute with $c$ , hence so does $c' = a^{- 1} (a c')$ . Thus $c$ commutes with all of $K^{*}$ , trivially with $0$ , and by Step 1 with all of $D ∖ K$ . Therefore $c \in Z (D)$ , and $K \subseteq Z (D)$ .

Corollary(13.18)Conjugates of a noncentral element generate

Let $D$ be a division ring and $d \in D ∖ Z (D)$ . Then $D$ is generated as a division ring by the set of conjugates ${x d x^{- 1} : x \in D^{*}}$ .

Proof

Let $K$ be the division subring generated by all conjugates of $d$ . For $x \in D^{*}$ , the division subring $x^{- 1} K x$ contains every $x^{- 1} (y d y^{- 1}) x = (x^{- 1} y) d (x^{- 1} y)^{- 1}$ , i.e. every conjugate of $d$ ; hence $K \subseteq x^{- 1} K x$ , that is $x K x^{- 1} \subseteq K$ . So $K$ is normal in $D$ . But $d \in K$ is noncentral, so $K not \subseteq Z (D)$ , and $(13.17)$ forces $K = D$ .

Corollary(13.19)Commutators generate

A noncommutative division ring $D$ is generated as a division ring by its multiplicative commutators $x^{- 1} y^{- 1} x y$ $(x, y \in D^{*})$ . The set of commutators is stable under every automorphism, so the division subring it generates is normal; it is not central because not all commutators are central when $D$ is noncommutative. See Multiplicative Commutators in Division Rings for the details and for the additive contrast.

Theorem(13.10)No small centralizers

Let $D$ be an infinite division ring with centre $F = Z (D)$ . Then for every $a \in D$ the field $F (a)$ is contained in an infinite subfield $K$ of $D$ . In particular $C_{D} (a) \supseteq K$ is infinite for every $a \in D$ .

Proof

$F (a)$ is a field by $(13.17 b)$ . If it is infinite, take $K = F (a)$ . So assume $F (a)$ is finite; then $F$ is finite, $char D = p > 0$ , and $D \neq F$ since $D$ is infinite. If $a \in F$ , replace $a$ by any element of $D ∖ F$ : proving the claim for that element also proves it for $a$ , since the resulting $K$ contains $F = F (a)$ . So assume in addition $a \notin F$ .

Now $a$ is noncentral and torsion in $D^{*}$ (it lies in the finite field $F (a)$ ), so Herstein's Lemma $(13.8)$ supplies $y \in D^{*}$ with $y a y^{- 1} = a^{i} \neq a$ for some $i > 0$ . Since $a^{i} \in ⟨ a ⟩$ , conjugation by $y$ normalises the finite cyclic group $⟨ a ⟩$ , giving a homomorphism $⟨ y ⟩ \to Aut (⟨ a ⟩)$ into a finite group; hence $y^{n}$ centralises $a$ for some $n \geq 1$ .

The element $y$ has infinite order. Indeed, if $y$ were torsion then $⟨ a ⟩ ⟨ y ⟩$ would be a finite subgroup of $D^{*}$ (as $y$ normalises $⟨ a ⟩$ ), hence cyclic by $(13.3)$ because $char D = p > 0$ , hence abelian — contradicting $y a y^{- 1} \neq a$ .

Finally $a$ , $y^{n}$ and $F$ commute pairwise, so $K : = F (a, y^{n})$ is a field by $(13.17 b)$ . It contains $F (a)$ , it contains the infinite-order element $y^{n}$ , hence is infinite, and every element of $K$ commutes with $a$ , so $K \subseteq C_{D} (a)$ .

Remark(13.17c)Faith's strengthening

Cartan–Brauer–Hua says a noncentral proper division subring $K ⊊ D$ cannot have $K^{*}$ normal in $D^{*}$ — that is, $N_{D^{*}} (K^{*}) \neq D^{*}$ . Faith proved much more: for such a $K$ the normaliser $N_{D^{*}} (K^{*})$ must have infinite index in $D^{*}$ . So the failure of normality is not marginal; the conjugates of $K^{*}$ form an infinite family.

Proof Techniques and Method

How these proofs work, and which move to reuse.

Move 1

Perturb by 1

Replace $a$ by $b = a - 1$ . The commutator with $c$ survives, and $a = b + 1$ collapses a product. This is the standard way to manufacture a second, related conjugator in a division ring.

Move 2

Solve for the outsider

Rearrange the identity so the element you want to exclude appears alone on one side. Membership of the other side in $K$ then contradicts $a \notin K$ .

Move 3

Insiders via outsiders

From ‘every outsider commutes with $c$ ’, get ‘every insider does too’ using $c' = a^{- 1} (a c')$ with $a$ and $a c'$ both outside $K$ . Costs one line, converts a partial statement into centrality.

Move 3 is worth isolating because it is what makes the theorem clean. Without it one only learns that $K$ and its complement commute, which is a statement about a partition rather than about the centre. The trick works because $K^{*}$ acts on $D ∖ K$ by multiplication without fixed points.

For $(13.18)$ and $(13.19)$ the reusable pattern is different: turn a conjugation-stable set into a normal division subring, then let $(13.17)$ decide. Any set $S$ with $x S x^{- 1} \subseteq S$ for all $x \in D^{*}$ generates a normal division subring, so the only question left is whether $S$ contains a noncentral element.

Worked Example

$ℂ$ inside the real quaternions

Take $D = ℍ$ with $Z (ℍ) = ℝ$ , and $K = ℂ = ℝ \oplus ℝ i$ . Here $K$ is a proper division subring and $K not \subseteq Z (D)$ , so $(13.17)$ predicts $K$ is not normal. Conjugation by $j$ is deceptive: $j i j^{- 1} = - i \in ℂ$ , and indeed $j z j^{- 1} = \bar{z}$ for all $z \in ℂ$ , so $ℂ$ is stable under that one inner automorphism.

Normality must be tested against all of $D^{*}$ . Put $a = 1 + j$ , with $a^{- 1} = (1 - j) / 2$ . Then

a^{- 1} i a = \frac{1}{2} (1 - j) i (1 + j) = \frac{1}{2} (1 - j) (i + k) = \frac{1}{2} (i + k + k - i) = k \notin ℂ,

(E.1)

Using $i j = k$ , $j i = - k$ , $j k = i$ .

so $ℂ$ is not normal in $ℍ$ , as the theorem demands.

Watching the proof solve for $a$

Run the proof of $(13.17)$ on this data: $a = 1 + j \notin ℂ$ , $c = i \in ℂ$ , $b = a - 1 = j$ . We computed $a^{- 1} c a = k$ , and $b^{- 1} c b = (- j) i j = k j = - i$ . The identity predicts

a (a^{- 1} c a - b^{- 1} c b) = (1 + j) (k - (- i)) = (1 + j) (k + i) = k + i + j k + j i = k + i + i - k = 2 i,

(E.2)

c - b^{- 1} c b = i - (- i) = 2 i . The two sides agree.

(E.3)

Solving as in $(13.13')$ : $(k + i)^{- 1} = - (k + i) / 2$ , so $a = 2 i \cdot (- (k + i) / 2) = - i (k + i) = - i k - i^{2} = j + 1$ . The identity reconstructs $a = 1 + j$ exactly — and it built it out of $i$ , $k = a^{- 1} i a$ and $- i = b^{- 1} i b$ , all of which would have had to lie in $K$ had $K$ been normal.

Comparison and Classification

The additive and multiplicative theorems of §13 side by side
Question	Additive version	Multiplicative version
Which elements are tested?	$a b - b a$	$x^{- 1} y^{- 1} x y$
Commuting with all of them forces centrality	$(13.4)$	$(13.15)$
All of them central forces $D$ commutative	$(13.5)$	$(13.16)$
They generate $D$	$(13.6)$ , but only together with $Z (D)$	$(13.19)$ , on their own
Subring theorem	$(13.7)$ : Lie ideal $K$ with $char K \neq 2$ is central	$(13.17)$ : normal $K \neq D$ is central
Characteristic hypothesis	needed — fails at $char 2$ as stated	none

Which hypotheses each result actually consumes
	(13.17)	(13.18)	(13.19)	(13.10)
$D$ a division ring	yes	yes	yes	yes
$K$ a division subring	yes	no	no	no
Stability under inner automorphisms	yes	derived	derived	no
$D$ noncommutative	no	implied by $d$ noncentral	yes	no
$D$ infinite	no	no	no	yes
Positive characteristic	no	no	no	only inside the proof

Which hypotheses each result actually consumes

Relationship Map

The theorem is a hub: one statement, several descendants, and one external input (Herstein's Lemma) used only for the companion result $(13.10)$ .

Identity $(13.13)$ — $a (a^{- 1} c a - b^{- 1} c b) = c - b^{- 1} c b$
- $(13.17)$ Cartan–Brauer–Hua
  - $(13.18)$ conjugates of a noncentral element generate $D$
  - $(13.19)$ multiplicative commutators generate $D$
  - Faith: $N_{D^{*}} (K^{*})$ has infinite index
- $(13.14)$ divided form
  - $(13.15)$ centralizing all commutators forces centrality
  - $(13.20)$ the upper central series of $D^{*}$ stops at once

(13.8)

Herstein's Lemma

(13.3)

finite subgroups cyclic in char

p

(13.10)

infinite subfield containing

F (a)

C_{D} (a)

infinite

The chain matters downstream: $(13.10)$ is the reason a maximal subfield of an infinite division ring cannot be finite, which is where Maximal Subfields of Division Rings begins.

Applications and Industry Use

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

Galois theory of skew fields

Why normality was abandoned

Henri Cartan's 1947 paper set up a Galois correspondence for noncommutative fields. The theorem shows the naive normality condition selects only central subfields, so the correspondence must be phrased with groups of automorphisms and inner-automorphism-free conditions instead.

Group theory of

D^{*}

Subnormal subgroup structure

The theorem is the base case for a large literature on normal and subnormal subgroups of $D^{*}$ : no proper noncentral division subring contributes one, so the interesting normal subgroups are not of subring type.

Division algebras

Rigidity of subalgebra lattices

In central simple algebra theory the corollary $(13.18)$ is used to show that a nonzero two-sided ideal or a conjugation-stable subalgebra is forced to be everything — the same argument pattern that underlies simplicity proofs.

Computer algebra

Invariance tests

For an explicitly presented division algebra (quaternion algebras and cyclic algebras in Magma, Sage or GAP), testing whether a subalgebra is conjugation-stable is a finite linear-algebra computation, and the theorem tells you in advance which answers are possible.

The honest summary: this is a structural rigidity theorem used inside algebra. Its engineering relevance is indirect — it is part of the reason division-algebra constructions used in space–time coding and in error-correcting codes have such tightly constrained subobject lattices.

Design Considerations

Design considerations here means the choices made when modelling a problem with these algebraic structures.

Do not model a skew field by analogy with groups. Quotients by normal division subrings do not exist as a useful construction, because the only candidates are central; use central simple algebra theory and Brauer groups instead.
Choose your invariant subobject carefully. If you need a subring stable under conjugation and noncentral, you are asking for $D$ itself. If you need a proper subring, expect the normaliser to be small — Faith's theorem quantifies how small.
**Test invariance on all of $D^{*}$ .** As the quaternion example shows, stability under conjugation by a handful of elements says nothing; $ℂ \subseteq ℍ$ is stable under conjugation by $i$ , $j$ and $k$ separately yet is not normal.
**Decide early whether $D$ may be commutative.** Both corollaries are vacuous or false-sounding for fields: $(13.18)$ needs a noncentral element to exist, and $(13.19)$ needs $D$ noncommutative.

Standards and Notation

Standards here covers notation, symbol and markup standards, and reference implementations, rather than material or design codes.

Multiplicative group

D^{*}

(Lam);

D^{\times}

is equally common and preferred in ISO 80000-2 style

Normality

K^{*} ⊴ D^{*}

; some authors write ‘

K

is invariant in

D

’

Centralizer

C_{D} (S)

here;

Z_{D} (S)

and

{Cent}_{D} (S)

appear in the literature

Centre

Z (D)

, occasionally

K (D)

in older French sources

NameCartan–Brauer–Hua; the Cartan is Henri Cartan, not Élie

ImplementationsMagma QuaternionAlgebra, Sage QuaternionAlgebra(QQ,a,b), GAP AlgebraByStructureConstants

Failure Modes and Common Mistakes

CounterexampleExercise 13.9Amitsur: the theorem fails for simple rings

Let $F = ℚ (x)$ and let $A = F [t; δ]$ be the differential polynomial ring, with $δ$ the formal derivative and multiplication determined by $t g (x) = g (x) t + δ (g (x))$ . Then $A$ is a simple domain, $U (A) = F^{*}$ (degrees add, so units are the nonzero constants), and hence $F = U (A) \cup {0}$ is carried to itself by every automorphism of $A$ — yet $Z (A) = ℚ$ , so $F not \subseteq Z (A)$ and $F \neq A$ .

So the trichotomy of $(13.17)$ genuinely uses that $D$ is a division ring, not merely a simple ring. See Differential Polynomial Rings and the Weyl Algebra for the simplicity of $A$ .

Do not drop the hypothesis $K \neq D$ : the conclusion would be false for $K = D$ whenever $D$ is noncommutative.
Do not read $(13.10)$ as saying $C_{D} (a)$ is a field. It says $C_{D} (a)$ contains an infinite field; the centralizer of a noncentral element in $ℍ$ is $ℂ$ , but centralizers are generally only division subrings.
Do not expect $(13.10)$ without infiniteness: in a finite division ring — a finite field, by Wedderburn — every centralizer is finite for the trivial reason.
The proof of $(13.10)$ silently uses that a finite subgroup of $D^{*}$ is cyclic in characteristic $p$ ; that fails in characteristic $0$ , where $ℍ^{*}$ contains the quaternion group of order 8.

Historical Notes and Lessons Learned

1905Wedderburn's Little TheoremFinite division rings are commutative — the first rigidity theorem of the subject, and the reason infiniteness appears as a hypothesis in $(13.10)$ rather than as an accident.
1947Henri CartanIn his paper on Galois theory for noncommutative fields, Cartan proves the theorem in the course of showing that the naive normality condition is useless for a Galois correspondence.
1949Richard BrauerBrauer publishes a short note, ‘On a theorem of H. Cartan’, giving an independent and shorter argument.
1949Hua Loo-KengHua obtains the same result independently while studying the multiplicative group of a skew field, in the same programme that produced the theorem that a solvable $D^{*}$ forces commutativity.
1950sFaith's quantitative formFor a noncentral proper division subring $K$ , the normaliser of $K^{*}$ is shown to have infinite index in $D^{*}$ — normality does not merely fail, it fails everywhere.
1955AmitsurAmitsur classifies the finite subgroups of division rings and produces the simple-ring example showing the theorem cannot be transplanted to simple rings.

The lesson worth keeping is that three people found the same one-line identity within two years. That is usually a sign the statement is the right one: it is what the algebra of $a$ , $a - 1$ and conjugation forces, and any programme that needed proper normal division subrings was doomed before it started.

Quick Reference

Theorem

K ⊊ D

division subring,

K^{*} ⊴ D^{*}

\Rightarrow

K \subseteq Z (D)

Engine

a (a^{- 1} c a - b^{- 1} c b) = c - b^{- 1} c b

with

b = a - 1

Corollary (13.18)Conjugates of a noncentral

d

generate

D

Corollary (13.19)Multiplicative commutators generate a noncommutative

D

Companion (13.10)

D

infinite

\Rightarrow

F (a)

lies in an infinite subfield;

C_{D} (a)

is infinite

Fails forSimple rings — Amitsur's

ℚ (x) [t; δ]

SharpeningFaith:

[D^{*} : N_{D^{*}} (K^{*})] = \infty

for noncentral

K ⊊ D

No hypothesis oncharacteristic, dimension, cardinality

Decision table for a division subring $K \subseteq D$
What you know about $K$	What follows
$K^{} ⊴ D^{}$ and $K \neq D$	$K \subseteq Z (D)$
$K^{} ⊴ D^{}$ and $K not \subseteq Z (D)$	$K = D$
$K$ contains a noncentral element and all its conjugates	$K = D$
$K ⊊ D$ and $K not \subseteq Z (D)$	$N_{D^{}} (K^{})$ has infinite index
$K = C_{D} (a)$ with $D$ infinite	$K$ is infinite

Frequently Asked Questions

Which Cartan is the Cartan of Cartan–Brauer–Hua?

Henri Cartan, in his 1947 work on Galois theory for noncommutative fields. Élie Cartan's name attaches to Lie theory results elsewhere; the confusion is common enough that Brauer's follow-up note is titled ‘On a theorem of H. Cartan’.

Why is there no hypothesis on the characteristic, when the additive analogue (13.7) needs char ≠ 2?

The additive proof adds two expressions to reach $2 (a^{2} c - a c a)$ , and that factor of $2$ has to be invertible. The multiplicative identity $(13.13)$ produces the element $a$ directly by a division, with no integer coefficient anywhere, so nothing can vanish in characteristic 2.

Does the theorem say anything about normal subgroups of the multiplicative group in general?

Only about those of the form $K ∖ {0}$ for a division subring $K$ . Plenty of other normal subgroups exist and are noncentral — the commutator subgroup $[D^{*}, D^{*}]$ for a start. The general subgroup structure of $D^{*}$ is a separate subject, treated on The Multiplicative Group of a Division Ring.

Is normality really equivalent to

x K x^{- 1} = K

for all

x

Yes. The stated condition $x K x^{- 1} \subseteq K$ applied to $x^{- 1}$ gives $x^{- 1} K x \subseteq K$ , i.e. $K \subseteq x K x^{- 1}$ . Both inclusions give equality, so no generality is lost by stating only the containment.

Why does the proof of (13.10) need Herstein's Lemma at all?

The only obstruction is a finite $F (a)$ sitting in an infinite $D$ . Herstein's Lemma is what converts ‘ $a$ is noncentral and torsion in characteristic $p$ ’ into a conjugator $y$ with $y a y^{- 1} = a^{i} \neq a$ , and a counting argument then forces $y$ to have infinite order — which is where the required infinite field comes from.

Can the theorem be extended to simple rings or to matrix rings?

Not verbatim. Amitsur's example $ℚ (x) [t; δ]$ is a simple domain in which the noncentral subfield $ℚ (x)$ is invariant under all automorphisms. There are valid generalisations to simple and semisimple rings, but each carries extra hypotheses — typically on units or on the invariant subring being a division ring.

References

T. Y. Lam, A First Course in Noncommutative Rings, Graduate Texts in Mathematics 131, Springer-Verlag, 1991, §13, especially (13.10), (13.13) and (13.17)–(13.19).
H. Cartan, “Théorie de Galois pour les corps non commutatifs”, Annales scientifiques de l'École Normale Supérieure 64 (1947).
R. Brauer, “On a theorem of H. Cartan”, Bulletin of the American Mathematical Society 55 (1949).
L. K. Hua, “Some properties of a sfield”, Proceedings of the National Academy of Sciences of the U.S.A. 35 (1949).
P. M. Cohn, Skew Fields: Theory of General Division Rings, Encyclopedia of Mathematics and its Applications 57, Cambridge University Press, 1995.
I. N. Herstein, Noncommutative Rings, Carus Mathematical Monographs 15, Mathematical Association of America, 1968, Chapter 3.

AI Suggested Questions

Write out the proof of the additive analogue (13.7) and show exactly where characteristic 2 breaks it.
Prove Faith's theorem that the normaliser of a noncentral proper division subring has infinite index.
Which normal subgroups of $ℍ^{*}$ exist, and how does that square with Cartan–Brauer–Hua?
Give a version of Cartan–Brauer–Hua for simple artinian rings and identify the extra hypotheses it needs.
How is (13.10) used to prove that maximal subfields of an infinite division ring are infinite?
Compare the conjugation-stability argument in (13.18) with the proof that a simple ring has no proper invariant ideals.
What is known about division subrings invariant only under conjugation by a fixed subgroup of $D^{*}$ ?

01Executive Summary

02Overview

03Learning Objectives

04Definitions

05Core Concepts

The identity that does all the work

Why the substitution b=a−1

From elementwise commuting to central

06Key Results

07Proof Techniques and Method

Perturb by 1

Solve for the outsider

Insiders via outsiders

08Worked Example

ℂ inside the real quaternions

Watching the proof solve for a

09Comparison and Classification

10Relationship Map

11Applications and Industry Use

Why normality was abandoned

Subnormal subgroup structure

Rigidity of subalgebra lattices

Invariance tests

12Design Considerations

13Standards and Notation

14Failure Modes and Common Mistakes

15Historical Notes and Lessons Learned

16Quick Reference

17Frequently Asked Questions

18Related KEVOS Topics

19References

20AI Suggested Questions

Continue learning

Executive Summary

Overview

Learning Objectives

Definitions

Core Concepts

Why the substitution $b = a - 1$

Key Results

Proof Techniques and Method

Worked Example

$ℂ$ inside the real quaternions

Watching the proof solve for $a$

Comparison and Classification

Relationship Map

Applications and Industry Use

Design Considerations

Standards and Notation

Failure Modes and Common Mistakes

Historical Notes and Lessons Learned

Quick Reference

Frequently Asked Questions

Related KEVOS Topics

References

AI Suggested Questions