To better understand it, you can do some calculations on smaller finite fields, for example using modulo 67. You can start from 1 and multiply it by two. You will get 1,2,4,8,16,32,64 and then suddenly you will get 128-67=61. Here you can see that dividing 61 by two will give you 64.
your calculation is wrong, 61 divided by 2 in this group does give you 64. read my post above.
your mistake is that you are calculating 61/2=30.5 using simple arithmetic whereas all the calculations here are modular arithmetic meaning:
61/2 ≡ 61 * 34 ≡ 2047 ≡ 64 (mod 67)
where 34 is calculated by solving this:
2*x ≡ 1 (mod 67)