Algebraically Closed Subfields | KEVOS® Mathematics

Q: Why can a noncommutative division ring not be finite-dimensional over an algebraically closed *central* subfield?

If K were central and algebraically closed with _K D < ∈fty, then every a ∈ D would be algebraic over K, hence in K, giving D = K and commutativity. (15.9) handles the harder situation where K is algebraically closed but not central: then K is a maximal subfield and the centre lies strictly below it.

Q: Where exactly is the Artin–Schreier theorem used?

Twice. In (15.9) it converts 1 < _F K < ∈fty into F real closed with _F K = 2, which is what forces the dimension 4. In (15.10) it is used contrapositively, to rule out the possibility that the given real closed field is already a maximal subfield.

Q: Does (15.10) imply that R is isomorphic to the centre?

No, and Lam's example shows why: there are real closed subfields R of C with R(-1) = C that are not isomorphic to R. Inside H_R such an R is a noncentral real closed subfield with (D_R) = 4. The conclusion controls only the joint configuration R(i) = F(i).

Q: Why does Noether–Jacobson need the algebra to be algebraic over F?

Separability is a statement about minimal polynomials, so an element must be algebraic before the question makes sense. Without algebraicity there may be no nonconstant polynomial relations at all — the free field over F, for instance, has elements transcendental over F.

Q: Is the Noether–Jacobson theorem constructive?

Not as proved here: the argument is a contradiction obtained from assuming pure inseparability everywhere, so it certifies existence without exhibiting an element. In the centrally finite case one can be more explicit, since the separable elements form a Zariski-dense set once one exists.

Q: Do these theorems say anything in characteristic p?

(15.9) and (15.10) are vacuous there — a real closed field has characteristic 0, and Artin–Schreier shows the configuration cannot occur in characteristic p. (15.11), by contrast, is interesting precisely in characteristic p, since in characteristic 0 it is immediate.

Executive Summary

Suppose a noncommutative division ring $D$ contains an algebraically closed field $K$ with $\dim (D_{K}) < \infty$ . Then $K$ has to be a maximal subfield, so the machinery of maximal subfields applies and $\dim_{F} D = r^{2}$ with $r = \dim_{F} K$ for $F = Z (D)$ . The Artin–Schreier theorem now takes over: a field of finite codimension greater than one inside an algebraically closed field is real closed of codimension exactly two. Hence $r = 2$ , $\dim_{F} D = 4$ , and Frobenius identifies $D$ as the quaternion algebra over $F$ .

Gerstenhaber and Yang weakened algebraically closed to real closed with the same conclusion; the argument first shows that the real closed subfield cannot itself be maximal, then upgrades it by adjoining $\sqrt{- 1}$ . The section closes with the Noether–Jacobson theorem, which supplies separable elements in any noncommutative algebraic division algebra.

$2$ Forced degree

r

$4$ Forced

\dim_{F} D

Real closedForced centre

QuaternionsForced isomorphism type

Overview

Frobenius' theorem says the only finite-dimensional division algebras over $ℝ$ are $ℝ$ , $ℂ$ and $ℍ$ . The results on this page are the same statement freed from $ℝ$ : what matters is not the real numbers but the Artin–Schreier configuration — a field sitting just below an algebraically closed field.

The mechanism is short. Finite codimension over an algebraically closed subfield $K$ kills the possibility of enlarging $K$ , because any commuting element would be algebraic over $K$ and hence already in $K$ . So $K$ is maximal, and Maximal Subfields of Division Rings gives $\dim_{F} D = (\dim_{F} K)^{2}$ . Since $K$ is algebraically closed and $F$ has finite codimension in it, Artin–Schreier leaves only one possibility.

K algebraically closed, \dim (D_{K}) < \infty, D \neq Z (D) ⟹ \dim_{Z (D)} D = 4 .

(15.9)

The dimension is forced before the isomorphism type is identified; Frobenius then does the rest.

The Artin–Schreier theorem is quoted, not proved here: if $K$ is algebraically closed and $F \subseteq K$ satisfies $1 < \dim_{F} K < \infty$ , then $F$ is real closed and $K = F (\sqrt{- 1})$ , so $\dim_{F} K = 2$ . In characteristic zero this is comparatively easy; the hard part of the general proof is showing that the characteristic must be zero.

Learning Objectives

Show that an algebraically closed subfield of finite codimension in a division ring is self-centralizing.
Combine $(15.8)$ with Artin–Schreier to force $r = 2$ .
State $(15.9)$ and $(15.10)$ with full hypotheses and distinguish the two.
Explain why the real closed subfield in $(15.10)$ need not be isomorphic to the centre.
State the Noether–Jacobson theorem $(15.11)$ and identify the role of algebraicity.
Reconstruct Herstein's proof using the inner derivation $δ_{a}$ and its nilpotence in characteristic $p$ .

Definitions

Real closed field: A field $F$ that is not algebraically closed but for which $F (\sqrt{- 1})$ is. Such an $F$ carries a unique ordering, its positive elements are exactly the squares, and every odd-degree polynomial over it has a root.
Artin–Schreier configuration: A pair $F \subseteq K$ with $K$ algebraically closed and $1 < \dim_{F} K < \infty$ . The theorem says $F$ must be real closed and $\dim_{F} K = 2$ .
Quaternion algebra over $F$: The $4$ -dimensional $F$ -algebra with basis $1, i, j, k$ , relations $i^{2} = j^{2} = - 1$ , $i j = - j i = k$ . Over a real closed $F$ it is a division algebra.
Separable element: An element $a$ algebraic over $F$ whose minimal polynomial over $F$ has no repeated roots. Over a perfect field — in particular in characteristic zero — every algebraic element is separable.
$δ_{a}$: The inner derivation $x \mapsto a x - x a$ of $D$ . It is the difference of the commuting operators of left and right multiplication by $a$ , so in characteristic $p$ one has $δ_{a}^{p^{n}} = δ_{a^{p^{n}}}$ .

Throughout, a subfield $K$ of $D$ is not assumed to contain $Z (D)$ in advance; when $K$ turns out to be maximal, $(15.7)$ supplies that containment.

Core Concepts

Algebraic closure forces maximality

Let $c \in C_{D} (K)$ . Since $\dim (D_{K}) < \infty$ , the powers $1, c, c^{2}, \dots$ are right $K$ -dependent, and because $c$ commutes with $K$ the dependence relation is an honest polynomial equation $\sum_{i} c^{i} a_{i} = 0$ with $a_{i} \in K$ . So $c$ is algebraic over $K$ ; as $K$ is algebraically closed, $c \in K$ . Hence $C_{D} (K) = K$ and $(15.7)$ makes $K$ a maximal subfield.

\dim (D_{K}) < \infty

every

c \in C_{D} (K)

algebraic over

K

C_{D} (K) = K

K

maximal

Where the number two comes from

By $(15.8)$ , $\dim_{F} K = r$ and $\dim_{F} D = r^{2}$ . Noncommutativity gives $D \neq F$ , so $r > 1$ . Now $F \subseteq K$ with $K$ algebraically closed and $1 < \dim_{F} K < \infty$ : this is exactly the Artin–Schreier configuration, and the theorem returns $F$ real closed with $K = F (\sqrt{- 1})$ , hence $r = 2$ and $\dim_{F} D = 4$ .

Why the real closed version needs an extra step

If the given subfield $R$ is merely real closed, it might a priori be maximal already. Ruling that out is the only new work: if $R$ were maximal then $(15.8)$ would give $\dim_{F} R = \dim (D_{R}) = s > 1$ , so $F$ would have codimension $2 s > 2$ in the algebraically closed field $R (\sqrt{- 1})$ — impossible by Artin–Schreier. Therefore a maximal subfield $K \supseteq R$ is strictly larger, is algebraic over $R$ , and so equals $R (\sqrt{- 1})$ , which is algebraically closed. Then $(15.9)$ applies.

Separability from a nilpotent derivation

The Noether–Jacobson theorem has a different flavour. Assume, for contradiction, that every element outside $F$ is purely inseparable. Then for $a \notin Z (D)$ some $p$ -th power $a^{p^{n}}$ is central, which makes the inner derivation $δ_{a}$ nilpotent. Nilpotence lets one descend to the last nonzero iterate, and a short manipulation produces an element $v$ with $v = 1 + a^{- 1} v a$ . Raising to a suitable $p$ -th power turns this into $v^{p^{m}} = 1 + v^{p^{m}}$ , i.e. $0 = 1$ .

Key Results

TheoremArtin–SchreierQuoted input

If $K$ is an algebraically closed field and $F$ is a subfield with $1 < \dim_{F} K < \infty$ , then $F$ is real closed, $K = F (\sqrt{- 1})$ and $\dim_{F} K = 2$ . (In particular $char F = 0$ .) A proof is in Jacobson, Basic Algebra II; the characteristic-zero case is substantially easier and is all that is needed for $(15.10)$ .

Theorem(15.9)Division rings containing an algebraically closed field

Let $D$ be a noncommutative division ring containing an algebraically closed field $K$ such that $r : = \dim (D_{K}) < \infty$ . Then $F : = Z (D)$ is a real closed field, $K = F (\sqrt{- 1})$ , $\dim_{F} D = 4$ , and $D$ is the division ring of quaternions over $F$ .

Proof

**Step 1: $K$ is a maximal subfield.** If $c \in C_{D} (K)$ then $c$ is algebraic over $K$ , because $\dim (D_{K}) < \infty$ makes the powers of $c$ right $K$ -dependent, and $c$ commutes with $K$ so the relation is polynomial. Algebraic closure gives $c \in K$ , so $C_{D} (K) = K$ and $K$ is maximal by $(15.7)$ . In particular $F = Z (D) \subseteq K$ .

Step 2: dimensions. By $(15.8)$ , $\dim_{F} K = \dim (D_{K}) = r$ and $\dim_{F} D = r^{2}$ . Since $D$ is noncommutative, $D \neq F$ and therefore $r > 1$ .

Step 3: Artin–Schreier. We have $F \subseteq K$ with $K$ algebraically closed and $1 < \dim_{F} K = r < \infty$ . Hence $F$ is real closed, $K = F (\sqrt{- 1})$ and $r = 2$ ; consequently $\dim_{F} D = 4$ .

Step 4: Frobenius. $D$ is a $4$ -dimensional noncommutative division algebra over the real closed field $F$ . By Frobenius' theorem in the form valid over any real closed field — the finite-dimensional case of the classification recorded in Lam §13 — the only finite-dimensional division algebras over $F$ are $F$ , $F (\sqrt{- 1})$ and the quaternion algebra; dimension and noncommutativity leave only the last. Hence $D ≅ (\frac{- 1, - 1}{F})$ .

Theorem(15.10)Gerstenhaber–Yang

Let $D$ be a noncommutative division ring containing a real closed field $R$ such that $s : = \dim (D_{R}) < \infty$ . Then $F : = Z (D)$ is a real closed field and $D$ is the division ring of quaternions over $F$ . Moreover there exists $i \in C_{D} (R)$ with $i^{2} = - 1$ such that $R (i) = F (i)$ inside $D$ — although $R$ and $F$ need not be isomorphic, and $R$ need not be central.

Proof

Let $K$ be a maximal subfield of $D$ containing $R$ ; one exists by Zorn's Lemma. We claim $K \neq R$ . Suppose $K = R$ . Then $\dim (D_{K}) = s$ , and $s > 1$ because $D$ is noncommutative. By $(15.8)$ , $s = \dim_{F} K = \dim_{F} R$ where $F = Z (D) \subseteq R$ . Consequently $F$ has codimension $\dim_{F} R (\sqrt{- 1}) = 2 s > 2$ in the algebraically closed field $R (\sqrt{- 1})$ , contradicting Artin–Schreier, which permits codimension $2$ only. (Only the characteristic-zero case of Artin–Schreier is used here, since a real closed field has characteristic $0$ .)

Hence $R ⊊ K$ . Since $\dim (D_{R}) < \infty$ , the field $K$ is a finite algebraic extension of $R$ ; as $R$ is real closed its only nontrivial algebraic extension is $R (i)$ with $i^{2} = - 1$ , so $K = R (i)$ and $K$ is algebraically closed. Also $i \in K = C_{D} (K) \subseteq C_{D} (R)$ .

Now $\dim (D_{K}) \leq \dim (D_{R}) < \infty$ , so $(15.9)$ applies to the algebraically closed subfield $K$ : $F = Z (D)$ is real closed, $D$ is the quaternion algebra over $F$ , and $K = F (\sqrt{- 1}) = F (i)$ . Combining, $R (i) = K = F (i)$ .

Example—

R

and

F

really can differ

Subfields of $ℂ$ maximal among those not containing $i$ are real closed with $R (\sqrt{- 1}) = ℂ$ , and among them there are fields not isomorphic to $ℝ$ . Inside $D = ℍ_{ℝ}$ we then have $R \subseteq ℂ \subseteq D$ with $\dim (D_{R}) = \dim_{ℂ} D \cdot \dim_{R} ℂ = 2 \cdot 2 = 4 < \infty$ , while $R$ is noncentral and not isomorphic to $Z (D) = ℝ$ .

Theorem(15.11)Noether–Jacobson

Let $D$ be a noncommutative division ring which is an algebraic algebra over a field $F \subseteq Z (D)$ — $F$ need not be all of $Z (D)$ . Then there exists an element of $D ∖ F$ that is separable over $F$ .

Proof

If $char F = 0$ every algebraic element is separable, and $D ∖ F \neq \emptyset$ because $D$ is noncommutative while $F$ is not. So assume $char F = p > 0$ and suppose, for contradiction, that every element of $D ∖ F$ is purely inseparable over $F$ .

Fix $a \in D ∖ Z (D)$ ; then $a \notin F$ , so $a^{p^{n}} \in F$ for some $n$ . Let $δ = δ_{a} : D \to D$ , $δ (x) = a x - x a$ . Left and right multiplication by $a$ commute as operators, so in characteristic $p$ we may take $p^{n}$ -th powers termwise: $δ^{p^{n}} (x) = a^{p^{n}} x - x a^{p^{n}} = 0$ , because $a^{p^{n}} \in F \subseteq Z (D)$ . Thus $δ$ is nilpotent, and $δ \neq 0$ since $a \notin Z (D)$ .

Choose $x$ with $δ (x) \neq 0$ and let $r \geq 1$ be largest with $y : = δ^{r} (x) \neq 0$ . Then $δ (y) = 0$ , i.e. $y$ commutes with $a$ ; hence so does $z : = y^{- 1} a$ . Writing $y = δ (u)$ with $u = δ^{r - 1} (x)$ and using $a z = z a$ ,

a = y z = (a u - u a) z = a (u z) - (u z) a = δ (v), v : = u z .

If $v \in F$ then $δ (v) = 0 = a$ , contradicting $a \neq 0$ ; so $v \in D ∖ F$ and by assumption $v^{p^{m}} \in F$ for some $m$ . From $a = a v - v a$ , left multiplication by $a^{- 1}$ gives $1 = v - a^{- 1} v a$ , that is $v = 1 + a^{- 1} v a$ . Raising to the $p^{m}$ -th power — legitimate because $1$ is central — yields

v^{p^{m}} = (1 + a^{- 1} v a)^{p^{m}} = 1 + a^{- 1} v^{p^{m}} a = 1 + v^{p^{m}},

the last equality because $v^{p^{m}} \in F$ is central. Hence $0 = 1$ , a contradiction. Therefore some element of $D ∖ F$ is separable over $F$ .

Proof Techniques and Method

The reusable moves behind these proofs.

Move 1

Finite codimension makes centralizers algebraic

If $\dim (D_{K}) < \infty$ then everything commuting with $K$ is algebraic over $K$ . Combine with a closure property of $K$ — algebraically closed, or real closed — to pin the centralizer down.

Move 2

Import a field-theoretic classification

Artin–Schreier and Frobenius are both statements about fields and real closed fields; the ring theory reduces the problem until one of them applies verbatim. Recognising the configuration is the entire skill.

Move 3

Use nilpotence of $δ_{a}$

In characteristic $p$ , $a^{p^{n}}$ central makes $δ_{a}$ nilpotent; passing to the last nonzero iterate produces an element killed by $δ_{a}$ , which can then be inverted and fed back. This is the standard route to separability statements.

Move 3 recurs throughout the theory of division rings — it is also how one proves that a noncommutative algebraic division algebra over a finite field cannot exist, and how the Cartan–Brauer–Hua theorem is approached.

Worked Example

The two hypotheses at work

A model case. $D = ℍ_{ℝ}$ contains $K = ℂ$ , algebraically closed, with $\dim (D_{K}) = 2 < \infty$ . Indeed $F = Z (D) = ℝ$ is real closed, $ℂ = ℝ (\sqrt{- 1})$ , $\dim_{F} D = 4$ and $D$ is the quaternion algebra — every clause of $(15.9)$ is visible at once.

**A model case that is not $ℝ$ .** Let $F_{0} = ℝ \cap \bar{ℚ}$ be the field of real algebraic numbers, which is real closed, and let $D_{0} = (\frac{- 1, - 1}{F_{0}})$ . Then $K_{0} = F_{0} (i) = \bar{ℚ}$ is algebraically closed, $\dim (D_{0})_{K_{0}} = 2$ , and $\dim_{F_{0}} D_{0} = 4$ . The theorem does not single out the real numbers; it singles out real closed fields, of which there are many non-isomorphic examples.

Dropping finite codimension

Let $k = ℂ (t)$ and let $σ$ be the automorphism $f (t) \mapsto f (t + 1)$ , of infinite order. Form the twisted Laurent series division ring $D = k ((x; σ))$ , with $x a = σ (a) x$ .

Commuting with $x$ forces each coefficient to lie in the fixed field $k^{σ} = ℂ$ ; commuting with $t$ forces $a_{i} \cdot i = 0$ , hence $a_{i} = 0$ for $i \neq 0$ since $char = 0$ . So $Z (D) = ℂ$ .
Thus $D$ is a noncommutative division ring containing the algebraically closed field $ℂ$ — but $\dim (D_{ℂ})$ is infinite, and $D$ is very far from quaternionic.
This is exactly the case excluded by the hypothesis $\dim (D_{K}) < \infty$ : without it, algebraically closed subfields are no constraint at all.

Comparison and Classification

Four theorems, one conclusion
Result	Hypothesis on $D$	Hypothesis on the subfield	Conclusion
Frobenius	finite-dimensional over $ℝ$	$ℝ$ central	$ℝ$ , $ℂ$ or $ℍ$
$(15.9)$	noncommutative division ring	$K$ algebraically closed, $\dim (D_{K}) < \infty$	$Z (D)$ real closed; $D$ quaternionic of dimension $4$
$(15.10)$	noncommutative division ring	$R$ real closed, $\dim (D_{R}) < \infty$	same, plus $R (i) = F (i)$
$(15.11)$	noncommutative, algebraic over $F$	$F \subseteq Z (D)$ , no closure assumed	a separable element exists outside $F$

What each hypothesis is responsible for
	$K$ maximal	$r = 2$	$D$ quaternionic	Separable element
$\dim (D_{K}) < \infty$	yes	yes	yes	no
$K$ algebraically closed	yes	yes	yes	no
$D$ noncommutative	no	yes	yes	yes
Artin–Schreier theorem	no	yes	yes	no
Frobenius theorem	no	no	yes	no
$D$ algebraic over $F$	no	no	no	yes

What each hypothesis is responsible for

Relationship Map

Two independent strands meet in this part of the section: the maximal-subfield machinery, and the classification of fields sitting below an algebraically closed field.

Maximal subfield machinery — $(15.7)$ and $(15.8)$
- Gives $\dim_{F} D = r^{2}$ and $\dim_{F} K = r$
- Applied to $K$ algebraically closed in $(15.9)$
- Applied to $K \supseteq R$ inside the proof of $(15.10)$
Field-theoretic classification — Artin–Schreier and Frobenius
- Artin–Schreier forces $r = 2$ and $F$ real closed
- Frobenius identifies the $4$ -dimensional algebra as quaternions
Separability — $(15.11)$ Noether–Jacobson
- Feeds the existence of separable maximal subfields, $(15.12)$
- Feeds Kaplansky's commutativity theorem via $(15.15)$

(15.8)

(15.9)

(15.10)

(15.12)

via

(15.11)

Applications and Industry Use

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

Mathematical physics

Why only three division algebras appear

The classification of finite-dimensional real division algebras underlies Dyson's threefold way in quantum mechanics: the commutant of an irreducible representation is $ℝ$ , $ℂ$ or $ℍ$ , which is why time-reversal symmetry classes come in exactly three flavours. $(15.9)$ is that classification with $ℝ$ replaced by an arbitrary real closed field.

Representation theory

Schur's lemma over non-closed fields

Over an algebraically closed field the endomorphism algebra of a simple module is the field itself; the results here explain what can happen one step below closure, and bound how badly Schur's lemma can fail.

Real algebraic geometry

Real closed fields as a working category

Model theory treats all real closed fields uniformly — they satisfy the same first-order sentences. $(15.10)$ is a ring-theoretic instance: the conclusion depends only on the Artin–Schreier configuration, not on the particular field.

Symbolic computation

Certifying a division algebra

When a computed algebra over a real field turns out $4$ -dimensional and noncommutative, $(15.9)$ removes any remaining case analysis: it is the quaternion algebra, and its multiplication table is determined.

Failure Modes and Common Mistakes

Do not assume $K \supseteq Z (D)$ as part of the hypothesis of $(15.9)$ — it is derived, via maximality.
Do not use the characteristic- $p$ part of Artin–Schreier casually: the theorem asserts that the configuration forces characteristic $0$ , which is the deepest part of its proof.
Do not expect an analogue of $(15.9)$ for simple artinian rings without adjustment: $M_{2} (ℂ)$ contains $ℂ$ with finite codimension and is not a division ring.
Do not conflate purely inseparable with inseparable: the proof of $(15.11)$ assumes the strong form, that every element outside $F$ has a $p$ -power in $F$ .

Historical Notes and Lessons Learned

1878FrobeniusClassifies finite-dimensional associative division algebras over $ℝ$ as $ℝ$ , $ℂ$ and $ℍ$ .
1927Artin and SchreierDevelop the theory of real closed and formally real fields, and prove that a proper subfield of finite index in an algebraically closed field is real closed of index two.
1930sNoether, JacobsonSeparable elements are shown to exist in centrally finite division algebras (Noether) and then in arbitrary algebraic division algebras (Jacobson), opening the way to separable maximal subfields.
1960Gerstenhaber and YangExtend Frobenius' theorem to division rings finite-dimensional over a real closed subfield, without assuming the subfield is central.

The methodological point: the classical theorems about $ℝ$ and $ℂ$ turn out to be theorems about an abstract configuration of fields. Once that is recognised, the ring theory becomes short and the field theory carries the weight.

Quick Reference

$(15.9)$ hypothesis

D

noncommutative,

K \subseteq D

algebraically closed,

\dim (D_{K}) < \infty

$(15.9)$ conclusion

Z (D)

real closed,

K = Z (D) (\sqrt{- 1})

\dim_{Z (D)} D = 4

D

quaternionic

$(15.10)$ hypothesis

D

noncommutative,

R \subseteq D

real closed,

\dim (D_{R}) < \infty

$(15.10)$ extra

\exists i \in C_{D} (R)

i^{2} = - 1

, with

R (i) = Z (D) (i)

Artin–Schreier

1 < \dim_{F} K < \infty

with

K

algebraically closed

\Rightarrow

F

real closed,

\dim_{F} K = 2

$(15.11)$

D

noncommutative and algebraic over

F \subseteq Z (D)

\Rightarrow

some element of

D ∖ F

is separable over

F

Key operator

δ_{a} (x) = a x - x a

; nilpotent when

a^{p^{n}} \in Z (D)

Reading the hypotheses
If you know	You may conclude	You may not conclude
$K$ algebraically closed, $\dim (D_{K}) < \infty$	$K$ is a maximal subfield	anything about $D$ if $D$ is commutative — then $D = K$
$D$ noncommutative as well	$\dim_{Z (D)} D = 4$ and $D$ is quaternionic	that $K ≅ ℂ$ or $Z (D) ≅ ℝ$
$R$ real closed, $\dim (D_{R}) < \infty$	$Z (D)$ real closed and $R (i) = Z (D) (i)$	that $R$ is central or isomorphic to $Z (D)$
$D$ algebraic over $F$ , noncommutative	a separable element exists outside $F$	that a separable maximal subfield exists

Frequently Asked Questions

Why can a noncommutative division ring not be finite-dimensional over an algebraically closed central subfield?

If $K$ were central and algebraically closed with $\dim_{K} D < \infty$ , then every $a \in D$ would be algebraic over $K$ , hence in $K$ , giving $D = K$ and commutativity. $(15.9)$ handles the harder situation where $K$ is algebraically closed but not central: then $K$ is a maximal subfield and the centre lies strictly below it.

Where exactly is the Artin–Schreier theorem used?

Twice. In $(15.9)$ it converts $1 < \dim_{F} K < \infty$ into $F$ real closed with $\dim_{F} K = 2$ , which is what forces the dimension $4$ . In $(15.10)$ it is used contrapositively, to rule out the possibility that the given real closed field is already a maximal subfield.

Does

(15.10)

imply that

R

is isomorphic to the centre?

No, and Lam's example shows why: there are real closed subfields $R$ of $ℂ$ with $R (\sqrt{- 1}) = ℂ$ that are not isomorphic to $ℝ$ . Inside $ℍ_{ℝ}$ such an $R$ is a noncentral real closed subfield with $\dim (D_{R}) = 4$ . The conclusion controls only the joint configuration $R (i) = F (i)$ .

Why does Noether–Jacobson need the algebra to be algebraic over

F

Separability is a statement about minimal polynomials, so an element must be algebraic before the question makes sense. Without algebraicity there may be no nonconstant polynomial relations at all — the free field over $F$ , for instance, has elements transcendental over $F$ .

Is the Noether–Jacobson theorem constructive?

Not as proved here: the argument is a contradiction obtained from assuming pure inseparability everywhere, so it certifies existence without exhibiting an element. In the centrally finite case one can be more explicit, since the separable elements form a Zariski-dense set once one exists.

Do these theorems say anything in characteristic

p

$(15.9)$ and $(15.10)$ are vacuous there — a real closed field has characteristic $0$ , and Artin–Schreier shows the configuration cannot occur in characteristic $p$ . $(15.11)$ , by contrast, is interesting precisely in characteristic $p$ , since in characteristic $0$ it is immediate.

References

T. Y. Lam, A First Course in Noncommutative Rings, Graduate Texts in Mathematics 131, Springer-Verlag, 1991, §15, results (15.9)–(15.11) (pp. 256–257).
T. Y. Lam, A First Course in Noncommutative Rings, §13 (Jacobson's theorem and the Frobenius-type classification over real closed fields).
N. Jacobson, Basic Algebra II, 2nd edition, W. H. Freeman, 1989 — the Artin–Schreier theorem on subfields of finite codimension in an algebraically closed field.
M. Gerstenhaber and C. T. Yang, “Division rings containing a real closed field”, Duke Mathematical Journal 27 (1960).
I. N. Herstein, Noncommutative Rings, Carus Mathematical Monographs 15, Mathematical Association of America, 1968, Chapter 3.
E. Artin and O. Schreier, “Algebraische Konstruktion reeller Körper”, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 5 (1927), 85–99.

AI Suggested Questions

Prove the characteristic-zero case of the Artin–Schreier theorem in detail.
Show that a finite-dimensional division algebra over a real closed field is isomorphic to $F$ , $F (\sqrt{- 1})$ or the quaternion algebra over $F$ .
Construct explicitly a real closed subfield $R$ of $ℂ$ with $R (i) = ℂ$ and $R$ not isomorphic to $ℝ$ .
Verify that the twisted Laurent series ring $ℂ (t) ((x; σ))$ with $σ (t) = t + 1$ is a division ring with centre $ℂ$ .
Give an example of an algebraic division algebra in characteristic $p$ with a purely inseparable maximal subfield, and locate the separable element promised by $(15.11)$ .
How does the nilpotence of the inner derivation $δ_{a}$ get used in the proof of the Cartan–Brauer–Hua theorem?
Is there a version of $(15.9)$ for simple artinian rings, and what replaces the quaternion conclusion?

01Executive Summary

02Overview

03Learning Objectives

04Definitions

05Core Concepts

Algebraic closure forces maximality

Where the number two comes from

Why the real closed version needs an extra step

Separability from a nilpotent derivation

06Key Results

07Proof Techniques and Method

Finite codimension makes centralizers algebraic

Import a field-theoretic classification

Use nilpotence of δa

08Worked Example

The two hypotheses at work

Dropping finite codimension

09Comparison and Classification

10Relationship Map

11Applications and Industry Use

Why only three division algebras appear

Schur's lemma over non-closed fields

Real closed fields as a working category

Certifying a division algebra

12Failure Modes and Common Mistakes

13Historical Notes and Lessons Learned

14Quick Reference

15Frequently Asked Questions

16Related KEVOS Topics

17References

18AI Suggested Questions

Continue learning

Executive Summary

Overview

Learning Objectives

Definitions

Core Concepts

Key Results

Proof Techniques and Method

Use nilpotence of $δ_{a}$

Worked Example

Comparison and Classification

Relationship Map

Applications and Industry Use

Failure Modes and Common Mistakes

Historical Notes and Lessons Learned

Quick Reference

Frequently Asked Questions

Related KEVOS Topics

References

AI Suggested Questions