5.4. Independence of characters.

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Theorem 5.13 (Dedekind’s theorem on the independence of characters). Let F be a

field, and let G be a group (monoid will do). Then any finite set {χ1, . . . , χm} of homomorphisms

G → F× is linearly independent over F, i.e.,

_

aiχi = 0 (as a function G → E) =⇒ a1 = 0, . . . , am = 0.

Proof. Induction on m. If m = 1, it’s obvious. Assume it for m − 1. We suppose

a1χ1(x) + a2χ2(x) + ・ ・ ・ + amχm(x) = 0 for all x ∈ G,

and show that this implies the ai to be zero. Since χ1 _= χ2, χ1(g) _= χ2(g) for some g ∈ G.

On replacing x with gx in the equation, we obtain the equation

a1χ1(g)χ1(x) + a2χ1(g)χ2(x) + ・ ・ ・ + amχ1(g)χm(x) = 0, all x ∈ G.

On multiplying the first equation by χ1(g) and subtracting it from the second, we obtain

the equation

a_

2χ2 + ・ ・ ・ + a_

mχm = 0, a_

i = ai(χi(g) − χ1(g)).

The induction hypothesis now shows that a_

i = 0 for all i ≥ 2. Since χ2(g) − χ1(g) _= 0, we

must have a2 = 0, and the induction hypothesis shows that all the remaining ai’s are also

zero.

6“Whenever n − 1i nvolves prime factors other than 2, we are always led to equations of higher degree....

WE CAN SHOW WITH ALL RIGOR THAT THESE HIGHER-DEGREE EQUATIONS CANNOT

BE AVOIDED IN ANY WAY NOR CAN THEY BE REDUCED TO LOWER-DEGREE EQUATIONS. The

limits of the present work exclude this demonstration here, but we issue this warning lest anyone attempt

to achieve geometric constructions for sections other than the ones suggested by our theory...and so spend

his time uselessly.”

42 J.S. MILNE

Corollary 5.14. Let F1 and F2 be fields, and let σ1, ..., σm be distinct homomorphisms

F1 → F2. Then σ1, ..., σm are linearly independent over F2.

Proof. Apply the theorem to χi = σi|F×

1 .

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