What are the requirements for such an endeavour? I am making a novel where the protagonist is reincarnated into a medieval society. His plan is to push forward technology into the modern age. He starts out as an Earl by the way
I made a slightly snarky comment earlier on the question, but I decided it deserves a more complete answer. You aren’t just wanting the transistor. You want the tech that enabled it. The problem is that the transistor is useless without a lot of other techs around it. For “radio and computer”, you need a hell of a lot. You need mining tech, which requires advances in architecture to build tunnels deep. You need smithing. You need gears. You need an understanding on light to build a monitor. You need math advanced enough to break down a frequency wave. You need power generation. And so on. The radio, specifically, was nearly simultaneously invented independently in many spots around the world because it was the next obvious tech after everything else that came before it.
In David Weber’s Safehold series, he “builds a transistor in a medieval setting” by taking 9 novels to build modern electronics up through every tech needed along the way. Takes about a century of in-book time. https://en.m.wikipedia.org/wiki/Safehold
Even something far simpler isn’t easy... look at William Gibson’s “The Difference Engine”.
My point is that building a transistor is in theory easy with materials, but practically impossible without industrial setting to give you the tools, math, electrical generation, and motivation to build it. It isn’t plausible unless you upend your medieval setting OR embed your medevial setting inside an industrial society, like a renaissance festival scenario.
Thinking too far Ahead
You are basically trying to invent a better flashlight battery before light bulbs were invented. Look into early wireless telegraph transmitters.
For a power source utilize a simple wet-cell voltaic pile, a primitive sort of battery. Its just zinc, copper, and salt water soaked felt disks piled up in alternating order. It would produce enough electricity to power a primitive spark-gap transmitter.
You will need alternating current instead of the direct current from a voltaic pile, so an inverter is needed. This is how simple a primitive inverter is.
For a transmitter You'd use a very large voltaic pile and a spark gap transmitter. It basically produces a very large band radio pulse every time an electrical ark of alternating current is jumped between two contacts which are attached to an antenna. It cannot transmit voices or audio, it simply is either transmitting or not, which can be used for wireless transmission of Morse code.
Boom, you now have people transmitting wireless Morse code 1000 years ahead of schedule and all you need is copper, zinc, saltwater, and wood.
I think rather than transistors, you're looking for a mechanical nand gate. There's a related question here. Nand gates are a logic gate that allows you to build complete computing systems, including adders and memory.
Combine these mechanical gates with some sort of mechanical clock (water wheel?) and you've got the capability to perform real computing.
Although not a direct answer to your question, you might want to check out "Connections" by James Burke. It's a documentary on the history of science and technology connecting the beginnings of science and technology to what was the "modern" world when it was produced in 1978. Key points to the series surround the connections between discoveries and developments and how one leads to or depends on another.
More directly in answer to your question:
As already stated in other answers, a transistor by itself is useless; you need all sorts of other things to build functional circuits, a source of good quality electrical power (stable voltage/frequency and sufficient current capacity), all sorts of theory of operation of electricity, electrical and electronic components.
You need all sorts of "modern" materials - plastics, ceramics, glass and specific metal alloys free of impurities. Depending on the type of transistor, you may need germanium (how do you identify the ore and extract the desired metal?) or silicon (pure, properly crystalized) and then you need the facilities to properly "dope" them with the correct type and amount of dopant. And then, how do you package and test the result?
Basically, you need to walk the medieval world through all the stages of science and technology that connects us to those times, pretty much in that order. I assume by medieval you mean iron age with the ability to make some steel (sword making). That means improved steelmaking, taking alchemy into modern chemistry - with the ability to isolate many if not most of the elements on the periodic table, developing math to include calculus, working with basic electricity, building steam engines and using them to power electric generators, making precision tools and instruments, building tube-based circuits first in order to make the instruments needed to measure what your semiconductors are doing.
I've surely left out countless intermediate (and essential) steps, but I hope it illustrates the point.
You can make mechanical logic gates. No? You want electric? Ok...
Considering that you say the protagonist is an Earl, I would place the history around the year 1000, taking as reference the Earls of Lade. With that in mind...
We have the Baghdad Battery which can be made to work as a galvanic battery (even if they weren't used like that, we know that works and that technology to make them is very old). In addition, the construction of a Voltaic Pile (which TCAT117 also mentions) is simple, if you know what you are doing. If it were not because of Luigi Galvani's discoveries, the motivation for Volta to attempt to build his Pile would not be there.
As you probably know, building coils and electromagnet is simple too. As long, as you have copper wires and iron rods. Casting iron is old technology for a medieval setting. The ancient knew copper, and the technology to create chain mails - which exist since the fourth or fifth century - can be adapted to create copper wires.
Furthermore, Theophrastus was already aware of magnetite, its attractive power, and where to mine it. If you have coils and permanent magnets, you can make a generator.
Note: With the coils you would build transformers, if you need them.
For making the generator move...
Creating springs in our world would require a few more centuries, however, if the protagonist spend some time developing metallurgy - perhaps with the help of a blacksmith - they should be viable. Once you have them, you can build relays.
While, relays are not full-blown transistors, they are electronic switches, and you can create logic gates out of them.
There is another thing you need to create a computer: a clock. At the time clock work was not fully developed. Gears already existed (Aristotle gave us the earlier description, there is the Antikythera mechanism, and there are mold for bronze Gears found in China dated to early Jin dinasty), and if you know how pendulums work, you can create a device that opens and close a circuit creating rhythmic electrical pulses... which will serve as clock for a rudimentary computer.
Note: If you think a single person can’t design a working primitive computer… go check out Building a Modern Computer from First Principles.
Ah, for the memory you can use the principle of the relays and use electromagnets to flip circuits. Do not build flip flop out of relays (I do not know if that would work in practice, but if it does, it is very inefficient), all you need is a lever that would rest to one side or the other closing a different circuit in each case, and that you can move it with electromagnets.
The technology to make electric lamps at the time would be too inneficient (stick with fossil fuels, and flamable oils, sorry). But you can use electromagnets move objects to serve as indicators. In fact, your memory is rather large; you could label some levers and read it directly.
It is also possible to open and close valves. This will be useful if you want to supply or cut the flow of water.
I suppose a rudimentary step motor is possible too, for instance using a cam (addendum: a lever should work too) and electromagnets to make it turn back and forth, and then transmit the movement via a ratchet. You could use that step motor to turn a gear. For example for a numeric display, or for moving another machine or you could even build a mechanical ALU.
You can build electromagnetic pulse transmitters and receptors as TCAT117 describe them.
I assumed year 1000. However, for the 13th century the technology to make wires and springs would be there. If use that for your setting, then there is no need to spend extra time developing stuff.
It is not simply a matter of inventing the transistor. You must ask, "What will power it?"
If you can refine copper to make wire, have magnetite and, lacquer for insulation then you can get most of the way to repeating the work of Tesla using those things and wood. (edit: and glass for insulators)
This will get you as far as motors, generators, distributed electricity and radio - although I am sure the occasional wooden electric motor may catch fire.
This will launch the industrial age - without petroleum distillates.
To create the literal transistor, you need to refine semi-conductors like silicone, germanium, etc. but, I have often wondered after looking at the foxhole radio (the diagram is more explanatory than anything else on that page) if some rudimentary transistor would not be possible using a similar method used to create the diode and providing a charge to overcome the bias (if that would work?).
I noticed that a lot of people think you need high accuracy tools to build a transistor, you dont, not if you are building a type of transistor that barely saw the light of day.
The first transistors made, were designed to replace the vacuum tube, and were huge, crude, inefficient, and ugly. But much better then vacuum tubes.
You could make one by hand using blacksmithing tools, if you know what materials to use. The hard part is you would need electricity, advances in material sciences and chemical engineering, in order to understand how make these crude transistors.
But the hardest part of all? Designing the low level instruction set for your first 100 square meter computer would be a nightmare for one person to attempt. (Eniac was the brain child of two geniuses and a team of scientists)
Edit: I think some think I an referring to modern nanometer transistors made in integrated circuits and that i am claiming they are "easy" to make. I am not, as even with our practically pure silicon and germanium their are still tons of gates that fail due to impurity, hence why cpu companies sell cups that are practically identical to each other as different chips because they had to disable certain cores, due to gate failures.
But large centimeter size transistors are makeable by people in their backyard. You just need the right cookbook, a lot of time, and enough money to pay for your many, many failed attempts.
This is true of a lot chemistry though, because it's the recipe that is hard to discover, one the recipe is written down it becomes simple, hence why it took a genius to realize nitro glycerine could be stabilized by mixing with powdered shell or clay and some type of sorbent, but anyone who can read can easily figure out how to do it today.
As others have mentioned, a transistor isn't much use without the context surrounding it. Aside from the physical problems (precision engineering, proper materials), there's also an issue of purpose. If you're not doing complex math or managing large numbers of things (people, dollars, livestock, etc), then a transistor won't necessarily accomplish much.
Maybe consider a Water computer? I'm not sure if one has ever been made, but the principles of transistors can be applied to fluid systems as well, with a few tweaks. As the article says, though, this would work on the order of millihertz, not megahertz, and could very well be slower than human calculation. It's also only useful for mathematics - no way to use it in a radio, for instance.
One possible benefit is the ability to perform algorithms to remote places without training mathematicians. A skilled craftsman could create the computer and the locals could do routine maintenance, with neither understanding how the system actually works. A book of designs could describe multiple requirements, so an algorithm would only need to be designed a single time before being deployed around the kingdom. If you needed to calculate something like "optimal flexibility of a catapult with length X", locals could use the computer's results without needing to learn the math behind it.
That said, I'm having a hard time coming up with an example where on-the-fly computation would be a better than a simple table of numbers, and it's almost certainly more efficient to simply grab a bunch of peasant kids and teach them math over a decade.
Building semiconductors would be pretty impossible in the era you have described.
There are a few different sorts of computers that are simple mechanical devices. https://www.kickstarter.com/projects/871405126/turing-tumble-gaming-on-a-mechanical-computer
I had one of the following computers. It was programed by moving the position of straws. You used a shuttle action to make it work. https://en.wikipedia.org/wiki/Digi-Comp_I and https://boingboing.net/2006/02/21/mechanical-computer.html
Look here https://www.google.com/search?q=homemade+mechanical+computer&newwindow=1&client=opera&sa=X&biw=1098&bih=546&tbm=isch&source=iu&ictx=1&fir=FWSssgxA0NJkRM%253A%252CCzcCkGNbCgdh-M%252C_&usg=__6zxnuLtlTQeP4HMHoV3fyll76b0%3D&ved=0ahUKEwjageL9gJbbAhWOt1MKHYxQCsYQ9QEIMzAC#imgrc=FWSssgxA0NJkRM:
You may also perform logic with relays if you want it to be electric. Steam power could flip flop valves.