Tower Extensions

Central Question

How does the size of a field extension change when we adjoin several elements in stages?

Why This Matters

Building fields step by step exposes the multiplicative behaviour of dimension and makes larger constructions understandable through smaller ones.

Mathematical Notes

Build the biquadratic field in two steps

Set $K=\mathbb Q(\sqrt2)$ and $L=K(\sqrt3)$. We know $[K:\mathbb Q]=2$. To find the next degree, first test whether $\sqrt3\in K$.

If $\sqrt3=a+b\sqrt2$ with rational $a,b$, squaring gives

Therefore $ab=0$. The case $b=0$ would make 3 a rational square; the case $a=0$ would make $3/2$ a rational square. Both are impossible, for instance by comparing prime exponents in a squared reduced fraction. Hence $x^2-3$ remains irreducible over $K$, and $[L:K]=2$.

If $u_1,\ldots,u_m$ is an $F$-basis of $K$ and $v_1,\ldots,v_n$ a $K$-basis of $L$, the products $u_iv_j$ are an $F$-basis of $L$. Expand first over $K$, then over $F$, to prove spanning. Group a putative dependence by the $v_j$ to prove independence in two stages.

Here this gives

What can and cannot be multiplied

Adjoining two square roots does not always give degree four:

The second degree is one because $\sqrt8=2\sqrt2$. Multiply the degree of each adjunction over the field already constructed, not its degree over the original field.

The tower law also constrains intermediate fields: if $F\subseteq M\subseteq L$, then $[M:F]$ divides $[L:F]$. An extension of prime degree has no proper intermediate field, even when it is not Galois.

Connection to geometric impossibility

A ruler-and-compass construction repeatedly solves linear and quadratic equations, so its coordinates lie in a tower whose total degree is a power of two. The number $\sqrt[3]2$ has degree three and therefore cannot arise in such a tower: duplicating a cube is impossible with those tools.

Being of power-of-two degree is a necessary condition for constructibility, not a sufficient condition for an arbitrary algebraic number. The stronger requirement is containment in a tower of quadratic extensions.

Finally, the two generators need not be permanent: $\sqrt2+\sqrt3$ generates $L$ alone. The primitive element theorem generalizes this phenomenon to every finite separable extension.

Learning Prompt

Teach me extensions of extensions. Start with $\mathbb Q(\sqrt2,\sqrt3)$. Make me determine its elements, basis and degree. Then derive the tower law intuitively. I want to see how increasingly complicated number systems are constructed one algebraic element at a time.

Ideas

Examples

Questions

Connections

• Algebraic Numbers and Minimal Polynomials
• Splitting Fields
• Field Automorphisms
• Week overview

One Thing That Surprised Me

Further Rabbit Holes

Sources

Ian Stewart, Galois Theory, fifth edition, CRC Press, §§6.2–6.3; §§7.2–7.4. Publisher and edition details.