The normal basis theorem is a fundamental result in Galois theory. For infinite fields, textbooks and monographs usually refer to a proof given by Artin in 1948. For finite fields, a completely different argument is commonly used. We give two short proofs of the normal basis theorem which work without this distinction.They build on Dedekind’s theorem on the linear independence of Galois automorphisms, and on the Krull–Schmidt theorem. The rest is elementary linear algebra. Both proofs are inspired by but simpler than the one given by Deuring in 1932

Blessenohl, Dieter

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Note di Matematica 27, n. 1, 2007, 5–10.On the normal basis theoremDieter BlessenohlMathematisches Seminar, Christian-Albrechts-Universita¨t Kiel,Ludewig-Meyn-Str. 4, 24098 Kiel, Germanyblessenohl@math.uni-kiel.deReceived: 19/10/2005; accepted: 20/12/2005.Abstract. The normal basis theorem is a fundamental result in Galois theory. For infinitefields, textbooks and monographs usually refer to a proof given by Artin in 1948. For finitefields, a completely different argument is commonly used.We give two short proofs of the normal basis theorem which work without this distinction.They build on Dedekind’s theorem on the linear independence of Galois automorphisms, andon the Krull–Schmidt theorem. The rest is elementary linear algebra. Both proofs are inspiredby but simpler than the one given by Deuring in 1932.Keywords: field theory, Galois theory, normal basis theoremMSC 2000 classification: 12F10, 01A60, 12-03Dedicated to Prof. Wolfgang Gaschu¨tz on the occasion of his 85th birthdayIn what follows, L/K is a finite Galois extension, G is the Galois group ofL/K and n := |G| = dimK L. Hence L is a KG-module in a natural way.Normal Basis Theorem. L is a regular KG-module.More explicitly, there exists an element x ∈ L such that {xα |α ∈ G} is aK-basis of L. Any such element x is called normal in L/K.Eisenstein, in 1850, was the first to state this result [6] for a finite field Lwith prime field K. In the same year, Scho¨nemann gave a proof of Eisenstein’s“Lehrsatz” under the assumption that the degree of the extension be prime [10].Hensel established the result for finite fields in full generality in 1888, apparentlywithout knowledge of the work of his predecessors [8].For certain infinite fields E.Noether showed the existence of a normal basis in1932 [9]. Her work was extended to arbitrary finite Galois extensions of infinitefields by Artin [1]; see also [2]. E. Noether was actually interested in normalbases which are also integral bases of the ring of algebraic integers — a muchmore difficult problem. Dedekind considered such bases as to be very useful inalgebraic number fields in 1880 already1.1Letter to Frobenius from July 8, 1896; printed in Dedekind, Coll. Papers. Vieweg 1931,Vol. 2, p. 433.6 D. BlessenohlIn his proof of 1932, Deuring did not need to distinguish between the finiteand the infinite case [4]; see also [5]. This is the only proof of that kind, asfar as we know. We present here two proofs of the normal basis theorem whichalso work equally well for finite and for infinite fields. The first one is a simplerversion of Deuring’s approach.First Proof. The field L is an (L,K) bimodule and a (K,KG) bimodule.Hence L⊗K L is an (L,KG)-bimodule, with the action of L and KG given byl(a⊗ b) = la⊗ b,(a⊗ b)ϕ = (a⊗ b)(id⊗ ϕ)= a⊗ bϕfor all l, a, b ∈ L and ϕ ∈ KG. In particular, L ⊗K L is an LG-module. Notethat dimL(L⊗K L) = n.Let µ : L ⊗K L → L , a ⊗ b 7→ ab denote the linearization of the multipli-cation in L. We defineλα := (id⊗ α)µ : L⊗K L→ Lfor all α ∈ G. It is clear that λα is K-linear. In fact, for all l, a, b ∈ L and α ∈ G,we have[l(a⊗ b)]λα = (la)bα = l(abα) = l[(a⊗ b)λα],hence λα is L-linear, that is, an element of the dual space (L ⊗K L)∗. It is animmediate consequence of Dedekind’s independence theorem that{λα |α ∈ G} is an L-basis of (L⊗K L)∗. (1)To see this, it is enough to show that {λα |α ∈ G} is L-linearly independentsince dimL(L⊗K L)∗ = n.Suppose the coefficients lα ∈ L (α ∈ G) are so chosen that∑α lαλα = 0,then, in particular,0 =∑αlα(1⊗ b)λα =∑αlαbαfor all b ∈ L. Thus Dedekind’s result yields lα = 0 for all α ∈ G. This implies (1).Following [7], we defineΦ : L⊗K L→ LG , x 7→∑α∈Gxλα · α−1.This map is L-linear and injective (by (1)), thus an L-isomorphism from L⊗KLonto LG since both spaces have the same L-dimension. For all x ∈ L⊗K L andOn the normal basis theorem 7β ∈ G, we have(xβ)Φ =(∑α∈Gxλβα · α−1β−1)β = xΦβ.Therefore Φ is actually an isomorphism of LG-modules. It follows that, on theone hand,L⊗K L ∼= LG ∼= KG⊕ · · · ⊕KG︸ ︷︷ ︸nas KG-modules . (2)On the other hand, if {a1, . . . , an} is any K-basis of L, thenL⊗K L = (a1 ⊗ L)⊕ · · · ⊕ (an ⊗ L).This is aKG-direct decomposition of L⊗KL. Besides, ai⊗L is aKG-submoduleof L⊗K L isomorphic to L, for all i. We conclude:KG⊕ · · · ⊕KG︸ ︷︷ ︸n∼= L⊕ · · · ⊕ L︸ ︷︷ ︸nas KG-modules . (3)If M is a module and k is a positive integer, we writek ·M :=M ⊕ · · · ⊕M︸ ︷︷ ︸k.Now supposeKG = a1 · U1 ⊕ a2 · U2 ⊕ . . . and L = b1 · V1 ⊕ b2 · V2 ⊕ . . .are decompositions of KG and L, respectively, into mutually non-isomorphic in-decomposable KG-modules U1, U2, . . ., respectively V1, V2, . . .. Then (3) impliesna1 · U1 ⊕ na2 · U2 ⊕ . . . ∼= nb1 · V1 ⊕ nb2 · V2 ⊕ . . . as KG-modules.It now follows from the Krull–Schmidt theorem thatU1 ∼= V1, U2 ∼= V2, . . . and na1 = nb1, na2 = nb2, . . .after a suitable relabelling of summands. In particular, we may deduce thata1 = b1, a2 = b2, . . . and thereforeKG ∼= L as KG-modules.This completes our first proof of the normal basis theorem. QED8 D. BlessenohlThe maps λα, α ∈ G, are algebra epimorphisms from L ⊗K L onto L. Let{dα |α ∈ G} denote the L-Basis of L ⊗K L dual to {λα |α ∈ G}, and setIα := Ldα =⋂α 6=β∈G ker λβ for all α ∈ G. ThenL⊗K L =⊕α∈GIα .This is the decomposition of L ⊗K L into simple ideals. Deuring’s proof takesthis decomposition as a starting point. However, his line of reasoning that theGalois group G permutes regularly the direct summands is different from ourargument above. Deuring also uses the Krull–Schmidt theorem to conclude hisproof. This seems to be a subsequently inserted correction due to E. Noetherin [4], and the argument is now usually referred to as the Deuring–Noethertheorem (see, for instance, [3]).Note that, for α, β ∈ G,(dαdα)λβ = dλβα dλβα ={1 if α = β,0 if α 6= β.Therefore dαdα = dα and dα is the neutral element of the algebra Iα. Further-more, dΦα = α−1 and IΦα = Lα−1.In this context, there is a remarkable characterisation of the normal elementsin L/K due to Erez [7]. An element a ∈ L is normal in L/K if and only if theelement(1⊗ a)Φ =∑α∈Gaα · α−1is a unit in the algebra LG. This can be seen as follows. We havea normal in L/K ⇔ L =⊕αKaα⇔ L⊗K L =⊕αL⊗ aα⇔ L⊗K L = 〈1⊗ a〉LG⇔ LG = 〈(1⊗ a)Φ〉LG.The generators of the LG-module LG are exactly the units of the algebra LG,and the proof is complete.As a K-vector space L ⊗K L is isomorphic to EndKL. This suggests toemploy EndKL for a proof of the normal basis theorem.On the normal basis theorem 9Second Proof. For all a ∈ L, let a : L→ L , x 7→ ax denote left multipli-cation by a. The map from L to EndKL which sends a 7→ a is an embedding ofalgebras. We denote the image of L under this mapping by L. From Dedekind’sindependence theorem we getEndKL =⊕α∈GαL as an L-vector space. (4)We have aα = αaα for all a ∈ L and α ∈ G, henceαL = Lαfor all α ∈ G. Take anyK-basis {a1, . . . , an} of L. Then {a1, . . . , an} is aK-basisof L. It follows thatαL = Lα = (a1K ⊕ · · · ⊕ anK)α = a1αK ⊕ · · · ⊕ anαK.for all α ∈ G. In particular, the set {ajα | 1 ≤ j ≤ n, α ∈ G} is a K-basis ofEndKL. Furthermore, for each 1 ≤ j ≤ n, Mj := 〈ajα |α ∈ G〉K is a regularKG-module since (ajα)β = aj(αβ) for all α, β ∈ G. Thus we have shown thatEndKL = M1 ⊕ · · · ⊕Mn ∼= KG⊕ · · · ⊕KG︸ ︷︷ ︸nas KG-modules. (5)For ϕ ∈ L∗ and a ∈ L, we defineϕ ∗ a : L→ L, x 7→ xϕ · a.Then ϕ ∗ a lies in EndKL and ϕ ∗ (a+ b) = ϕ ∗ a+ϕ ∗ b for all ϕ ∈ L∗, a, b ∈ L,and furthermore(ϕ ∗ a)f = ϕ ∗ affor all f ∈ EndKL. In particular, ϕ ∗ L := {ϕ ∗ a | a ∈ L} is a right ideal ofEndKL. If ϕ 6= 0, then this right ideal has dimension 1 as an L-vector space,and it is isomorphic to L as a KG-module.Let {ϕ1, . . . , ϕn} denote the basis of L∗ dual to {a1, . . . , an}. If x = k1a1 +· · ·+ knan ∈ L with k1, . . . , kn ∈ K, then for all f ∈ EndKLxf = xϕ1af1 + · · ·+ xϕnafn,hencef = ϕ1 ∗ af1 + · · ·+ ϕn ∗ afn ∈ ϕ1 ∗ L+ · · ·+ ϕn ∗ L.It follows thatEndKL = ϕ1 ∗ L⊕ · · · ⊕ ϕn ∗ L ∼= L⊕ · · · ⊕ L︸ ︷︷ ︸nas KG-modules (6)since dimL EndKL = n. The claim now follows from (5) and (6) as in the firstproof, by means of the Krull–Schmidt theorem. QED10 D. BlessenohlBoth L⊗K L and EndKL have K-dimension n2. Hence they are isomorphicas K-vector spaces. But this is not a canonical isomorphism. However, there isa canonical isomorphism ϑL from L∗ ⊗K L onto EndKL which sendsϕ⊗ a 7→ (x 7→ xϕ · a)for all ϕ ∈ L∗ and a ∈ L. The image of ϕ ⊗ a under ϑL is ϕ ∗ a. This is theunderlying idea of the isomorphism (6).We finally remark that ϕ∗L is an irreducible right ideal of the algebra EndKLwhenever ϕ 6= 0. Therefore (6) is in fact a direct decomposition of EndKL intominimal right ideals.References[1] E. Artin: Linear Mappings and the Existence of a Normal Basis, Interscience Publ.,Volume for Courant’s 60th birthday, (1948), 1–5[2] E. Artin: Galoissche Theorie, Harri Deutsch Verlag, Zu¨rich, Frankfurt, 1973.[3] R. W. Curtis, I. Reiner: Representation Theory of Finite Groups and AssociativeAlgebras, Interscience Publ., New York, London, 1962.[4] M. Deuring: Galoissche Theorie und Darstellungstheorie, Mathematische Annalen, 107,(1932), 140–144.[5] M. Deuring: Algebren. Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer,Berlin, 1935.[6] G. Eisenstein: Lehrsa¨tze, J. reine angew. Math., 39, (1850), 180–182.[7] B. Erez: Galois modules and class field theory, Geometry and Topology Monographs,Vol. 3, Invitation to higher local fields, Part II, section 10, 2000.[8] K. Hensel: U¨ber die Darstellung der Zahlen eines Gattungsbereiches fu¨r einen beliebigenPrimdivisor, J. reine angew. Math., 103, (1888), 230–273.[9] E. Noether: Normalbasis bei Ko¨rpern ohne ho¨here Verzweigung. J. reine angew. Math.,167, (1932), 147–152.[10] T. Scho¨nemann: U¨ber einige von Herrn Dr. Eisenstein aufgestellte Lehrsa¨tze, J. reineangew. Math., 40 (1850), 185–187.

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On the normal basis theorem

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On the normal basis theorem

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