i need to find how many automorphisms are there for the field Q(sqrt2).(adsbygoogle = window.adsbygoogle || []).push({});

i am given the hint to use the fact that a is a root of a polynomial on Q(sqrt2).

what i did is:

if f:Q(sqrt2)->Q(sqrt2) is an automorphism then if P(x) is a polynomial on this field then:

if a is a root of P(x), then P(a)=0 but then also f(P(a))=f(0)=0

so [tex]P(x)=b_0+b_1x+...+b_nx^n[/tex]

and [tex]f(P(x))=f(b_0)+f(b_1)f(x)+....+f(b_n)f(x)^n[/tex]

they have the same degree, so when f(P(x))=0=P(x)

we have: [tex]f(b_0)+f(b_1)f(x)+...+f(b_n)f(x)^n=b_0+...+b_nx^n[/tex]

from here iv'e concluded that it has only the identity function as an automorphism, am i right?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Automorphisms of Q(sqrt2)

**Physics Forums | Science Articles, Homework Help, Discussion**