# Thread: Are these vector spaces isomorphs?

1. ## Are these vector spaces isomorphs?

Let K1={a + (2^0.5)*b} | a,b rational numbers}, and K2={a + (3^0.5)*b} | a,b rational numbers} be two fields with the common multiplication and addition. Isomorphs are the following vector spaces :
(Q^n ., +; K1) and (Q^n ., +; K2) ?

2. ## Re: Are these vector spaces isomorphs?

Originally Posted by gotmejerry
Let K1={a + (2^0.5)*b} | a,b rational numbers}, and K2={a + (3^0.5)*b} | a,b rational numbers} be two fields with the common multiplication and addition. Isomorphs are the following vector spaces :
(Q^n ., +; K1) and (Q^n ., +; K2) ?
If you are asking whether $\mathbb{Q}[\sqrt{2}]=\left\{a+b\sqrt{2}:a,b\in\mathbb{Q}\right\}$ and $\mathbb{Q}[\sqrt{3}]=\left\{a+b\sqrt{3}:a,b\in\mathbb{Q}\right\}$ are isomorphic as vector spaces over $\mathbb{Q}$, then I have only one question--what are the dimensions of each of these spaces.