Can the trident created via divine trident be thrown? Do you have to retrieve it or does it just materialize in your hand? Can other creatures wield it?
No it cannot be thrown
While I agree with @ruffdove's answer, it is worth noting that Divine Trident is simply an electricity version of the classic Flame Blade spell. The descriptive text makes it a "4-foot-long blazing forked bolt of electricity" instead of a "3-foot-long blazing beam of red hot fire", but the game mechanics are absolutely identical, even including the ignition of highly flammable items. Relevant characteristics are:
- Attacks with the blazing beam / blazing forked bolt are melee touch attacks, not ranged attacks or ranged touch attacks. If the spell could be used to make either melee or ranged attacks then this would be specified (eg Produce Flame).
- The beam "springs forth from your hand", it is not a created object that is grasped in your hand. If your hand is somehow able to be non-traumatically removed or extended (eg with the Long arm spell) then the blade springing from the hand can be used to make melee touch attacks from the hand's location.
With this context in mind, it is worth examining a couple of related questions here and here on Flame Blade, admittedly for 3.5 rather than PF. There are certainly some ambiguities in the wording for particular situations, but the ability to make attacks at range does not appear to be one of them.
No, the spell's range is zero feet. It's attached to your hand.
To establish what the description says:
A 4-foot-long, blazing, forked bolt of electricity springs forth from your hand. You wield this spear-like bolt as if it were a trident (you are considered proficient with the bolt). Attacks with the trident are melee touch attacks. The bolt deals 1d8 points of electricity damage + 1 point per 2 caster levels (maximum +10). Since the bolt is immaterial, your Strength modifier does not apply to the damage. The bolt can ignite combustible materials such as parchment, straw, dry sticks, and cloth.
It says you may wield the bolt as it were a trident, and you may certainly throw a trident (though with a 10' range increment, it isn't ideally used in this way). However, the description goes on to say that all attacks with the bolt are melee touch attacks, which strongly implies that it may not be thrown.
I think the deciding factor here is the fact that the bolt "springs forth from your hand" and the spell's range is 0'. Note it says "from your hand" and not "in your hand." This implies that it is attached to you and you are not holding it with the option to drop or throw. Further, the spell's range is 0', which means that the bolt may not break contact your hand.