The 1837 edition omitted the definite article the before half here in Alma 20:23. It was restored to the RLDS text in the 1908 RLDS edition. Usage elsewhere supports the use of the before half in the phrase “half of X”, as in the following sampling:
Besides the instances of “the half of the kingdom” in Alma 20:23 and in Ether 8:2, there are five other examples with the exact phraseology (all are found in Ether 8–11). So clearly, the use of the the here in Alma 20:23 is correct. The 1837 loss of the the in this phrase was probably unintentional since that phrase (“the half of the kingdom”) was left unchanged elsewhere in the 1837 edition.
In the original text, there are nine instances of “one half of X” (including two of “the one half of X”), one instance of “a half of X”, and one of “half of X” (the last of these is what we typically expect in current English). For discussion of the tendency to omit the a from “a half ” in the Book of Mormon text, see under Alma 11:15. Obviously, variation is possible. The critical text will, in each instance, follow the evidence from the earliest textual sources in assigning the determiner for “half of X”. Here in Alma 20:23, the original the will be restored.
Interestingly, the precise phrase “the half of the kingdom” occurs three times in the King James Bible, all in the book of Esther:
There is also a related example in Mark 6:23 that has the the: “whatsoever thou shalt ask of me I will give it thee / unto the half of my kingdom”. Thus the use of “the half of the kingdom” is undoubtedly correct in Alma 20:23.
Summary: Restore in Alma 20:23 the original the to the phrase “even to the half of the kingdom”.