Tower law

Given three fields, $F\subseteq K\subseteq L$ the tower law states that if $[L:K]$ and $[K:F]$ are both finite then so is $[L:F]$ and $[L:F] = [L:K][K:F].$ Furthermore, if either $[L:K]$ or $[K:F]$ is infinite then so is $[L:F]$ .

Proof

First consider the case where $[L:K]$ and $[K:F]$ are both finite. Let $[L:K] = m$ and $[K:F] = n$ . Let $\{\alpha_1,\alpha_2,\ldots,\alpha_m\}$ be a basis for $L$ over $K$ and $\{\beta_1,\beta_2,\ldots,\beta_n\}$ be a basis for $K$ over $F$ . We claim that the set $B=\{\alpha_i\beta_j|1\le i\le m, 1\le j\le n\}$ (which clearly has $mn$ elements) is a basis for $L$ over $F$ .

First we show that $B$ spans $L$ . Take any $v\in L$ . As $\{\alpha_1,\alpha_2,\ldots,\alpha_m\}$ is a basis for $L$ over $K$ , we can write $v = \sum_ia_i\alpha_i$ , where $a_1,a_2,\ldots,a_n\in K$ . And now as $\{\beta_1,\beta_2,\ldots,\beta_n\}$ is a basis for $K$ over $F$ we can write $a_i=\sum_jb_{ij}\beta_j$ where $b_{i1},b_{i2},\ldots,b_{in}\in F$ , for each $a_i$ . So now $v = \sum_ia_i\alpha_i = \sum_i\left(\sum_jb_{ij}\beta_j\right)\alpha_i = \sum_{i,j}b_{ij}\alpha_i\beta_j.$ So indeed $B$ spans $L$ over $F$ .

Now we show that $B$ is independent. Assume that there are some $\{b_{ij}\}\subseteq F$ such that $\sum_{i,j}b_{ij}\alpha_i\beta_j = 0$ . So then we have $\sum_i\left(\sum_jb_{ij}\beta_j\right)\alpha_i = 0.$ So, as $\{\alpha_1,\alpha_2,\ldots,\alpha_m\}$ is independent over $K$ we get that $\sum_jb_{ij}\beta_j = 0.$ For all $i$ . And hence as $\{\beta_1,\beta_2,\ldots,\beta_n\}$ is independent over $F$ we get $b_{ij}=0$ for all $i$ and $j$ . Therefore $B$ is indeed independent.

Therefore $B$ is indeed a basis, so $[L:F] = |B| = mn$ , as desired.

Now we consider the infinite case. By the above argument if $\{\alpha_1,\alpha_2,\ldots,\alpha_m\}\subseteq L$ is independent over $K$ and $\{\beta_1,\beta_2,\ldots,\beta_n\}\subseteq K$ is independent over $F$ then the set $B=\{\alpha_i\beta_j|1\le i\le m, 1\le j\le n\}$ is independent over $F$ . Hence if either of $[L:K]$ and $[K:F]$ is infinite then there exisit arbitrarily large independent sets in $L$ over $F$ , so $[L:F]$ is infinite as well.

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Tower law

Proof