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The Enterprise arrives at the planet Ekos, where the Federation historian, John Gill, was observing the native culture. Ekos is an industrialized planet, but does not yet have space flight, and is a rather warlike society. The neighboring planet, Zeon, is more advanced and has basic space flight. The Federation has lost contact with John Gill, so the Enterprise was sent to investigate. When the arrive in orbit, they are attacked by a primitive rocket-borne nuclear warhead. They manage to detonate it at a safe distance.
Kirk and Spock plan to beam down in casual native clothing, which is an unusual but sensible tactic. They also have subdermal transponders implanted under the skin in their arms; with these transponders, Scotty will be able to locate them and beam them up, even without communicator contact. This is a wonderful idea that makes so much sense - it's too bad we never see it again in the series. With such a technology available, I might have even expected all Starfleet personnel to be required to wear one continually. Think about how many times Kirk or the others were abducted or lost contact with the Enterprise, and the transponders could have really helped.
When they beam down, they quickly discover that the Ekosian society has been transformed into a Nazi regime. They watch Nazi soldiers beat a Zeon man visiting Ekos. Then they see a public broadcast, which features John Gill as the fuhrer. They knock out two Nazi soldiers and swipe their uniforms so they can move more freely. But when they try to get into a headquarters building on their way to find John Gill, a high-ranking Nazi soldier realizes something is fishy about Spock's appearance (underneath his helmet) and when Spock's ears are revealed, they are both taken into custody.
They are whipped as Zeon spies to encourage them to talk. This is one of the most brutal scenes in the entire series, since Kirk and Spock are restrained and can't even evade the torture, let alone fight back. A high-ranking party official orders them confined to a cell for an hour, after which time they will be executed.
A Zeon named Isak is in the cell next to them, and he is more than happy to tell them about the recent history of Ekos. The Ekosians hadn't always had an antipathy toward Zeons, but in the past few years the rise to power of the Nazis also lead to the bigotry. This obviously fits in with John Gill coming to power as the fuhrer. Kirk is more determined than ever to find Gill.
In a laughable bit of engineering, Spock uses their transponders, metal from the bed frame, and the lightbulb to make a laser to cut through the lock on their cell door. This is so ridiculous - there's no way to collimate the laser light or to direct the beam. If the metal from the bed frame is available, why not use it to pick the lock instead? Plus, they've lost their backup option of being beamed up via the transponders.
They do escape, and they take Isak with them so he can show them where their communicators and phasers are being experimented with in the building's laboratory. They recover the pieces of their equipment, but they have already been partially disassembled and damaged, so Spock can only get one communicator functioning. They are discovered and are forced to flee, following Isak into the anti-Nazi underground.
The Zeons and Ekosians of the underground don't trust Kirk and Spock immediately, which is some nice plotting. All too often, native people trust Kirk for very little reason; the runaway slaves in "Bread and Circuses" come to mind here. The members of the underground even stage a raid by a Nazi officer, which involves Daras, a young Ekosian woman who was made a hero of the fatherland. She is actually against the Nazis, but puts on a loyal public face in order to spy on them from within.
Having gained the trust of the underground, Kirk convinces them that he and Spock need to reach John Gill and that if they do, they might be able to stop the Nazi machine. Frankly, the people of the underground don't seem nearly upset enough about the idea that a Federation person was responsible for this crisis in their society.
The fuhrer is going to be giving a speech that night, allegedly about the launch of their "final solution" to the Zeon problem: an attack against Zeon using Zeon rockets that they have captured. Kirk, Spock, and Isak finagle their way into the speech site by posing as a camera crew taping Daras attending the speech. They get a glimpse of Gill through a window in a door, and Kirk and Spock think he looks like he's drugged or otherwise mentally incapacitated. They call the Enterprise and have McCoy beam down in the uniform of a Nazi doctor.
John Gill begins his speech as the fuhrer, and it is composed of random and disconnected thoughts. It's clear to the Starfleet men and the underground that Gill is no longer dictating his own actions - his subordinate, Melakon, has taken over and is manipulating Gill and the Nazi government.
The Starfleet men manage to get into the room with Gill, and McCoy pronounces him drugged. He gives Gill something to begin to counteract the drug, but it will take a long time to get him completely sober. Kirk is able to question him about what happened. Gill said he wanted to calm the Ekosians' warlike nature by introducing the efficiency of the Nazi government, but didn't intend to include the bigotry and violence.
Kirk manages to get Gill functioning well enough to start a speech recalling the attack on Zeon and revealing Melakon as a traitor. Melakon mows down Gill with a machine gun, but Nazi soldiers that are members of the underground taken Melakon into custody. Now that the fuhrer is dead - after denouncing the war - and Melakon has been removed from power, but underground thinks they'll be able to move their society toward more peaceful goals. So the Enterprise leaves.
I cannot believe TOS made this episode only a generation after World War II. The episode's premise that some of the structure of Nazi society might be beneficial seems like a calculated insult to those who fought against the Nazis. And, frankly, the episode was not convincing about its premise. The efficiency and purpose of Nazi society were only possible with the zealotry inspired by the extreme bigotry espoused by Hitler. It's unbelievable that a life-long studier of history that John Gill was would not realize this.
The one good aspect of the episode was that it showed why the Prime Directive of the Federation was implemented. I do believe that a clever person with the advantage of Federation history and knowledge could manipulate events to put himself in charge. And then even though Gill had the best intentions when he started, events went quickly out of control. We can see how meddling with this society put them on the road to committing genocide. Those are pretty severe consequences, hence the Prime Directive's priority.