Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 18J (Galois Theory)

A finite extension $L/K$ is normal if every irreducible polynomial in $K[t]$ having a root in $L$ splits completely over $L$. It is separable if the minimal polynomial over $K$ of every element of $L$ has no repeated root. It is Galois if

Put $L_i=K({\alpha }_1,\dots ,{\alpha }_i)$, with $L_0=K$, and letFix a $K$-embedding $\sigma :L_{i-1}\to \overline{K}$. If $m_i(t)$ is the minimal polynomial of ${\alpha }_i$ over $L_{i-1}$, an extension of $\sigma$ to $L_i$ is determined by the image of ${\alpha }_i$, which must be a root of the polynomial obtained by applying $\sigma$ to the coefficients of $m_i$. Conversely, each distinct root gives one extension. Since $\overline{K}$ is algebraically closed, there is at least one such root, and there are at most $d_i$ distinct roots.

Starting from the unique embedding of $K$, induction therefore givesIf $L/K$ is separable, each $m_i$ is separable, so every transformed polynomial has exactly $d_i$ distinct roots. Every inequality is then an equality. Taking $i=n$ proveswith equality in the separable case.

After fixing $L\subseteq \overline{K}$, every $K$-automorphism of $L$ is a $K$-embedding $L\to \overline{K}$. HenceIf $L/K$ is normal, every $K$-embedding of $L$ into $\overline{K}$ has image $L$. It is consequently a $K$-automorphism of $L$, so the two sets, and therefore their cardinalities, are equal.

If $L/K$ is separable, part (a) givesIf it is also normal, part (b) givesThuswhich is exactly the defining condition for $L/K$ to be Galois.

Take $p=11$. Write $Y=X^5$, so the extension isThe polynomial $t^5-Y$ is irreducible over ${\mathbb{F}}_{11}(Y)$, for example by Eisenstein's criterion at $Y$ in ${\mathbb{F}}_{11}[Y]$, and has $X$ as a root. Thus the extension has degree five.

The element $3\in {\mathbb{F}}_{11}^{\times }$ has order five. Hence all roots of $t^5-Y$ areand already lie in ${\mathbb{F}}_{11}(X)$. The extension is therefore the splitting field of $t^5-Y$ and is normal. Its derivative is $5t^4\ne 0$ in characteristic eleven, so the polynomial is separable. By part (c),