names in a quadratic field extension

Question

Consider the quadratic field extension $\Bbb{Q}(\sqrt d)$.

1. Is there a good name to use for the parameter $d$?
2. Are there good names for $a$ and $b$ in the expression $a + b\sqrt d$?

For example, when $d = -1$ we call $a$ the real part and $b$ the imaginary part. Are there names which are used for a general quadratic extension?

Answer 1

Let $k=\mathbb{Q}(\sqrt{d})$.

1. If $d$ is non-zero, squarefree and congruent to $1$ modulo $4$, then $d$ is the discriminant of $k$ (if $d\not\equiv 1~(\operatorname{mod} 4)$ then $4d$ is the discriminant). Other than that, I'm not sure of any other standard name.

2. There is a canonical choice of power basis $\lbrace 1,\sqrt{d}\rbrace$ for $k$ as a vector space over $\mathbb{Q}$. Every element of $k$ can then be expressed as a vector of two rational numbers $\mathbf{x}=\binom{a}{b}$ corresponding to the element $a+b\sqrt{d}$. In this basis then, $a$ and $b$ are just the co-ordinates of the vector $\mathbf{x}$.