fix: proofread Galois and Ramification Theory part

[vEnhance]

This proofreads the Algebraic NT II: Galois and Ramification Theory part (tex/alg-NT/{galois,finite-field,ramification,frobenius,artin}.tex), corresponding to issue #315.

Most fixes are minor (typos, grammar, notation slips). A few are small but genuine mathematical/terminology errors, called out below.

Notable corrections

• galois.tex — The extension $\QQ(\sqrt2,\sqrt3)/\QQ$ was stated to be the splitting field of $(x^2-5)^2-6$. The minimal polynomial of $\sqrt2+\sqrt3$ is $(x^2-5)^2-24 = x^4-10x^2+1$ (since $(\sqrt2+\sqrt3)^2 = 5+2\sqrt6$). With $-6$ the roots are $\pm\sqrt{5\pm\sqrt6}$, generating a different field. Fixed $-6 \to -24$.
• ramification.tex — "$S_3$ has no subgroups of order $2$" is false ($S_3$ has three, generated by transpositions) and also contradicts the just-established $|D_{\kp_i}|=2$. The intended (and logically needed) statement is that $S_3$ has no normal subgroups of order 2. Fixed.
• artin.tex — Kronecker–Weber was stated as "$L$ is contained in a cyclic extension $\QQ(\zeta)$"; cyclotomic extensions are abelian but not generally cyclic (e.g. $\QQ(\zeta_8)$). Changed "cyclic" → "cyclotomic".
• artin.tex — Two places call the conductor $\kf$ the "discriminant"; the book reserves $\kf$ for the conductor (and even contrasts the two). Changed "discriminant" → "conductor".

Minor typos / grammar / notation

• galois.tex: $\QQ(\alpha_1,\dots,\alpha_n)$ → \alpha_m (matches the $\alpha_1,\dots,\alpha_m$ tower and $\tau_m$ in the diagram).
• finite-field.tex: Fibonacci numbers in a hint were typeset as \FF_p,\FF_{p+1},\dots (field notation) instead of F_p,F_{p+1},\dots; "lowest common denominator" → "least common multiple" (the next line uses \lcm); "elements ... is a quadratic residue" → "are quadratic residues"; "element $x$ in $p$" → "in $F$".
• frobenius.tex: "for every prime $p$" → "for every $\alpha \in \OO_K$" ($p$ is already fixed); missing subscript \Frob{\kp_2} → \Frob_{\kp_2}; Let $K = \Gal(\QQ(\zeta_m)/\QQ)$ → Let $K = \QQ(\zeta_m)$ (then $\Gal(K/\QQ)$ makes sense).
• ramification.tex: $K/F$ → $L/F$ in the general-extension remark (the surrounding text uses $\OO_L$ and "primes in $L$"); several "an unit"/"an UFD"/"an representative" → "a".
• artin.tex: stray \Frob_\kP(x) → \Frob_\kP; stray period inside a display; missing closing parenthesis; C_K → C_\QQ (and $P_K,I_K$) in a worked example whose base field is $\QQ$; H(L/\QQ, m) → H(L/\QQ, m\infty); \Gal(L/K)/L/K → E/K in the Hilbert class field solution (the field is named $E$).

No large arguments were rewritten.

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https://claude.ai/code/session_01FnhfNnwcsW5a9fNqNDPnu9

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