TITLE: A question about fields containing a copy of $\Bbb{Q}$
QUESTION [5 upvotes]: When we say a field contains a copy of the field of rational numbers $\Bbb{Q}$, what does this really mean? Does it mean it contains a field isomorphic to $\Bbb{Q}$, or does it mean it contains $\Bbb{Q}$ itself?
Also, say the field contains $\Bbb{Q}$. Consider the element $a+a+a=3a$. Here, does $3$ belong to $\Bbb{Q}$? Or is $3a$ just a way of representing $a+a+a$?
Thanks

REPLY [5 votes]: Does it mean it contains a field isomorphic to $\Bbb{Q}$, or does it mean it contains $\Bbb{Q}$ itself?
My answer to this would be "yes and yes."
The reason is that the whole idea of isomorphism means that we don't distinguish between isomorphic things. They have exactly the same properties, so they are the same. You can say they are different sets, but really it would be better to stop thinking of $\Bbb Q$ as some magical set that is different from its isomorphic counterparts, somehow.
For any ring with identity, there is always a homomorphism of $\Bbb Z$ Into the ring mapping 1 to the identity. If the characteristic is zero, then this map is an injection. If the ring your mapping into is a field with characteristic 0, this extends to a homomorphism from $\Bbb Q$ into your ring. All of the resulting images of elements of $\Bbb Q$ in your ring are exactly what you expect.