Note that these from linear algebra notes.
İt was defined fields, showed $\mathbb{Q}$ is a field. Then, below-mentioned qustion was proved. Yet, I didn't ask what happened. Can you explain? What deoes this question question?
Question: Find $a^ı, b^ı$ $\in$ $\mathbb{Q}$ such that $\dfrac {1} {a+b\sqrt {2}}= a^ı+b^ı\sqrt {2}$.
This is a question about field extensions. The idea is basically that we can think of adding $\sqrt{2}$ to the rationals in a consistent way. You can then show that {1,$\sqrt{2}$} is a basis for this new field when considered as a vector space over the rationals. So what is being asked is to find the coefficients of 1 and $\sqrt{2}$ such that they equal the inverse of $a + b\sqrt{2}$ for some fixed a and b in the rationals. Both these new coefficients should be rational numbers. This can be done pretty mechanically by finding some basic relationships between the elements of your new field.
Hint: Since $\sqrt{2}$ satisfies a quadratic polynomial over the rationals, I suggest you might see whether every element of your new field does, too. That'll give you a handle on what things need to look like.
Does that help?