Adjoining a Root

Central Question

How can we construct a number system in which $x^2-2$ has a root?

Why This Matters

A familiar irrational number becomes a first example of deliberately constructing a field, rather than merely enlarging a list of numbers.

Mathematical Notes

Construct the arithmetic, then identify the field

Introduce a symbol $\alpha$ subject to $\alpha^2=2$. Polynomial expressions reduce to $a+b\alpha$ with $a,b\in\mathbb Q$. Multiplication is forced:

This produces the ring $\mathbb Q[x]/(x^2-2)$; $\alpha$ means the class $[x]$, not a previously chosen real number. Each class has one representative of degree less than two.

To see that it is a field, take $a+b\alpha\ne0$:

The denominator cannot vanish unless $a=b=0$: otherwise $b\ne0$ would make $(a/b)^2=2$. Thus adjoining the root requires no additional rational expressions beyond these linear ones.

Evaluation $[f]\mapsto f(\sqrt2)$ identifies this constructed field with

The kernel of evaluation on $\mathbb Q[x]$ is $(x^2-2)$: polynomial division leaves a remainder $a+bx$, and $a+b\sqrt2=0$ forces both coefficients to vanish. This proves the identification rather than merely suggesting it.

A useful calculation

The operation $a+b\sqrt2\mapsto a-b\sqrt2$ preserves addition and multiplication. Its product with the original element is the rational number $a^2-2b^2$, a first encounter with the field norm.

Choosing $[x]\mapsto-\sqrt2$ gives an equally valid identification. The abstract construction distinguishes a generator, but its arithmetic does not privilege one of the two real roots.

Learning Prompt

I know undergraduate group, ring and field theory. Teach me what it really means to construct $\mathbb Q(\sqrt2)$, starting from $\mathbb Q$. Derive it rather than just defining it. Connect it to $\mathbb Q[x]/(x^2-2)$. Make me answer questions along the way. Focus on intuition and mathematical significance, not exam preparation.

Ideas

Examples

Questions

Connections

• Why Quotient by a Polynomial?
• Algebraic Numbers and Minimal Polynomials
• Week overview

One Thing That Surprised Me

Further Rabbit Holes

Sources

Ian Stewart, Galois Theory, fifth edition, CRC Press, §§4.1–4.3, 5.3–5.4. Publisher and edition details.