Torsion Points - Elliptic Curves: Number Theory and Cryptography - page 88

S = aT 1 + bT 2 for some integers a, b . Therefore,

e n ( S, dT 2 )= e n ( T 1 ,dT 2 ) a e n ( T 2 ,dT 2 ) b =1 .

Since this holds for all S , (2) implies that dT 2 = ∞ .Since dT 2 = ∞ if and
only if n|d , it follows that ζ is a primitive n th root of unity.

COROLLARY 3.11
If E [ n ] ⊆ E ( K ) ,then μ n ⊂ K .

R EM ARK 3.12 Recall that points in E [ n ] are allowed to have coordinates
in K . The hypothesis of the corollary is that these points all have coordinates
in K .

PROOF Let σ be any automorphism of K such that σ is the identity on
K .Let T 1 ,T 2 be a basis of E [ n ]. Since T 1 ,T 2 are assumed to have coordinates
in K ,wehave σT 1 = T 1 and σT 2 = T 2 .By(5),

ζ = e n ( T 1 ,T 2 )= e n ( σT 1 ,σT 2 )= σ ( e n ( T 1 ,T 2 )) = σ ( ζ ) .

The fundamental theorem of Galois theory says that if an element x ∈ K is

fixed by all such automorphisms σ ,then x K .Since ζ

is a primitive n th root of unity by Corollary 3.10, it follows that μ n ⊂

∈ K . Therefore, ζ ∈

K .
( Technical point: The fundamental theorem of Galois theory only implies
that ζ lies in a purely inseparable extension of K .Butan n th root of unity
generates a separable extension of K when the characteristic does not divide
n , so we conclude that ζ ∈ K .)

COROLLARY 3.13
Let E be an elliptic curve defined over Q .Then E [ n ]

⊆ E ( Q ) for n ≥ 3 .

PROOF If E [ n ] ⊆ E ( Q ), then μ n ⊂ Q , which is not the case when n ≥ 3.

REMARK 3.14 When n =2,itispossibletohave E [2] ⊆ E ( Q ). For
example, if E is given by y 2 = x ( x − 1)( x + 1), then

E [2] = {∞, (0 , 0) , (1 , 0) , ( − 1 , 0) }.

If n =3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 12, there are elliptic curves E defined over Q that
have points of order n with rational coordinates. However, the corollary says
that it is not possible for all points of order n to have rational coordinates for
these n . The torsion subgroups of elliptic curves over Q will be discussed in
Chapter 8.