Given a certain set of real numbers [math]S[/math].  Suppose for any two real numbers [math]a,b[/math] in [math]S[/math], [math]a+b[/math], [math]a-b[/math], [math]ab[/math] and [math]\frac{a}{b}[/math] (if [math]b[/math] is non-zero) are also in [math]S[/math], we say that [math]S[/math] is a [b][color=#0000ff]field[/color][/b].  Addition, subtraction, multiplication and division are collectively called [b][color=#0000ff]rational operations[/color][/b].[br][br][b]Examples[/b][br][list][*]The set of all rational numbers [math]\mathbb{Q}[/math] is obviously a field.[/*][*]The set of all constructible numbers [math]X [/math] is also field because we have already shown that addition, subtraction, multiplication and division can be done by Euclidean constructions, which implies that if [math]a[/math] and [math]b[/math] are two constructible numbers, [math]a+b[/math], [math]a-b[/math], [math]ab[/math] and [math]\frac{a}{b}[/math] (if [math]b[/math] is non-zero) are also constructible numbers.[/*][/list][br][br][br][br]A field [math]F[/math] is called [b][color=#0000ff]Euclidean[/color][/b] if for any positive real number [math]x [/math] in [math]F[/math], [math]\sqrt{x}[/math] is also in [math]F[/math].[br][br]The set of all constructible numbers [math]X[/math] is a Euclidean field because we can construct the square root of a length by Euclidean construction.[br][br]Next, we are going to investigate the underlying structure of [math]X[/math] in more detail.