2 Let Q 2 a b 2 ab Q Find all automorphisms of Q 2 no vec

2. Let Q[ 2] = {a + b 2 | a,b Q}. Find all automorphisms of Q[ 2]

no vector space please

Solution

An automorphism by definition is an isomorphism

Also Q[sqrt(2)] is a field because it satisfies all the properties of addition, multiplication, identities and inverse.

As an automorphism f: Q[ 2] ---> Q[ 2]

So what f does is f(2)= for some Q then you will know what f does to any element of Q(2) is because f(t)=t for all tQ and f is a ring homomorphism, hence

f(a+b2) = Some sort of c + d2 = c + d

So we need to figure out all the alphas

For auto morphism we must also have f(0) = f((2)^2 -2) = ^2 - 2 = 0

So for all alpha satisfying this type of equation, we have automorphisms

So alpha can be + or - sqrt(2)