Unique Factorization Domains

Lets consider the number 2. Looking for a factorization of 2 over $ latex \mathbb{Z}$ we have $2 \cdot 1$ . Is this factorizartion unique ? If we are looking over another domain like $latex \mathbb{Z}[\sqrt{-6}]$ , then how easy is to find all the posible factorizations of 2 ? If we find one is it indeed unique?

$2=(a+b\sqrt{-6})(c+d\sqrt{-6}) ,a,b,c,d \in \mathbb{Z}$ considering the norm of 2 : $N(2)=N((a+b\sqrt{-6}))(c+d\sqrt{-6})$ . But $N(2)=(2+0\sqrt{-6})(2-0\sqrt{-6})=4$ .Taking norms for both sides we have : $4=(a^2+6b^2)(c^2+6d^2)$ in $latex \mathbb{Z}$. On the RHS the factors can be a) either both of them 2 ,b) one of them 4 and the other 1.

a) It’s easy to so that the equation $2=x^2+6y^2$ has no integer solutions (just consider $(x,y)mod3$ ).

b) $1=x^2+6y^2 \rightarrow 1=1+06$ , which gives the factorization of norms: $4=4\cdot1$ which is trivial factorization on $latex \mathbb{Z\sqrt{-6}}$ (we are looking for non-trivial ones).

One can read a little more diophantine_eq and how the integer equation $2x^3=y^2+1$ can be solved over a unique factorization domain (UFD) such as $\mathbb{Z[i]}$ .

Galois #3

i wrote some of the very-very basics of field extensions , in fact the examples help to understand what an extension from a field to another is.field_extensions

galois#2

Consider the set of all the homorfisms of an extension of two fields $F \rightarrow E$ , so we can try to configure how the set of this homorfisms is related to the field or roots of a given polynomial , the number of this set with the degree of the extension .In generall if $Aut_{F}E = G(E,F)$ then $|G(E,F)| \le [E:F]$ , where $[E:F]$ is the degree of the extension from F to E.