English
The polynomial algebra F_q[X] embeds injectively into ringOfIntegers F_q F.
Русский
Алгебра многочленов F_q[X] инъективно включается в ringOfIntegers F_q F.
LaTeX
$$$\operatorname{algebraMap}_{F_q[X]} : F_q[X] \to \mathrm{ringOfIntegers}_{F_q}F \text{ is injective}.$$$
Lean4
theorem map_mul' (x y : RatFunc Fq) : inftyValuationDef Fq (x * y) = inftyValuationDef Fq x * inftyValuationDef Fq y :=
by
rw [inftyValuationDef, inftyValuationDef, inftyValuationDef]
by_cases hx : x = 0
· rw [hx, zero_mul, if_pos (Eq.refl _), zero_mul]
· by_cases hy : y = 0
· rw [hy, mul_zero, if_pos (Eq.refl _), mul_zero]
· simp_all [RatFunc.intDegree_mul]