How do I enchant my gear so I can cast spells for free?

I found out that you can enchant armor to reduce the Magicka cost of a certain magic school to zero. What are the requirements to do this, other than 100 Enchanting? Do I need certain perks to do this, or a potion?

I gave a detailed explanation of the formula in this answer.

net magnitude = base magnitude * soul multiplier * skill multiplier *
                (1 + Enchanter perk) * (1 + specific perk modifier)

where skill multiplier is "approximately"

skill multiplier = 1 + (skill / 100) * (skill / 100 - 0.14) / 3.4

See below, but here's a rundown:

• 25% on 4 items
  • 100 Enchanting
  • Perks (5 total)
    • Enchanter (4 ranks)
    • Insightful Enchanter
• 29% on 3 items + Archmage's Robes (recommended)
  • All of the above
  • 100 Alchemy
  • Perks (5 + 8 = 13 total)
    • Enchanter (5th rank)
    • Alchemist (5 ranks)
    • Physician
    • Benefactor
• 33% on 3 items
  • All of the above
  • 70 Restoration
  • Perks (13 + 3 = 16 total)
    • Novice Restoration
    • Regeneration
    • Necromage
  • Utilizing the Falmer Helmet + Circlet glitch for Fortify Alchemy

There are three main ways to use the 33%/Necromage build: maxing out schools, spreading the benefits evenly or using the Archmage's Robes. Again, I'd recommend the Archmage's Robes. Note that you can invest an additional 2 perk points in any of the above to get Extra Effect (via Corpus Enchanter). The magicka cost becomes as follows:

------------------------------------------------------------------------
  Option             Magicka Cost         (+ Extra Effect)    Perks Used
------------------------------------------------------------------------
  25% x 4             0/100/100/100/100    0/ 0/100/100/100      5 (7)
  29% x 3 + AR        0/ 85/ 85/ 85/ 85    0/ 0/ 85/ 85/ 85     13 (15)
* 33% x 4 + NM v1     0/ 59/100/100/100    0/ 0/ 18/100/100     16 (18)
* 33% x 4 + NM v2    18/ 18/100/100/100   18/18/ 18/ 18/100     16 (18)
* 33% x 3 + NM + AR   0/ 40/ 82/ 82/ 82    0/ 0/  0/ 82/ 82     16 (18)

* All of these assume you remain a vampire, in which case any Fortify School bonus will be increased by 25%.

To further increase the above effects, you can utilize the Falmer Helmet + Circlet exploit.

25% on 4 items

Since perks are more dear than skill points, let's assume you have 100 skill and an Enchanter's Elixir (alchemy produces weak potions even at 100 skill unless you invest some perk points):

skill multiplier = 1 + 1.25 * (1.25 - 0.14) / 3.4 = 1.4081

We can plug in some values to get to 25% net magnitude:

25% = 8% * 1 * 1.4081 * (1 + Enchanter) * (1 + Insightful Enchanter)
(1 + Enchanter) * (1 + Insightful Enchanter) = 2.2193

Once we have one rank in Enchanter, we can get Insightful Enchanter, which is the best single perk for Fortify School enchantments and provides a 25% bonus (plus you should probably go this route to get to Extra Effect). Therefore, we only need to solve for the number of ranks in Enchanter:

(1 + Enchanter) * 1.25 = 2.2193
Enchanter = 0.77

Therefore, at minimum we need 4 ranks in Enchanter.

29% on 3 items + Archmage's Robes

Another possible build uses the Archmage's Robes, which already provide 15% Fortify to all schools of magic. I prefer this because a) it's more attainable than Necromage, b) being a vampire really isn't that fun, and c) it doesn't require the Falmer Helmet exploit and d) you also get +50 magicka and +100% magicka regen. You just need the maxed alchemy/enchanting loop:

skill multiplier = 1 + (132 / 100) * (132 / 100 - 0.14) / 3.4
                 = 1.4581
net magnitude = 8% * 1 * 1.4581 * 2 * 1.25 = 0.2916

The requisites:

• 100 Alchemy
• 100 Enchanting
• Alchemist (all 5 ranks), Physician, Benefactor
• Enchanter (all 5 ranks), Insightful Enchanter

33% on 3 items

It goes without saying that we'll need all of the above and then some. Then we need to solve for what amount of bonus x we need to apply to our Enchanting skill to get to 34%:

33% = 8% * 1 * 2 * 1.25 * skill multiplier
skill multiplier = 1.65 = 1 + (1 + x) * (1 + x - 0.14) / 3.4
x = 0.5583

So you need at least a 56% boost to Enchanting. The solution to this is becoming a vampire with the Necromage perk. Necromage boosts the effect of everything you do to an undead by 25%. If you're undead, that includes you. So all of your own enchantments and potions are also 25% stronger. The alchemy/enchanting loop can then net you a +46% Fortify Enchanting potion. This requires:

• Maxed alchemy/enchanting loop
  • 100 Alchemy
  • 100 Enchanting
  • Alchemist (all 5 ranks), Physician, Benefactor
  • Enchanter (all 5 ranks), Insightful Enchanter
• Necromage
  • 70 Restoration
  • Novice Restoration, Regeneration, Necromage
• Misc
  • The Falmer Helmet/Circlet glitch, where you can enchant both with Fortify Alchemy and wear them simultaneously

You would use a fortify X enchantment, which you need to obtain by disenchanting an item with that enchantment on it already. It will reduce the cost of casting for that particular school of magic.

You can do 4 items at 25% each with a grand soul stone, with only 5 perks in enchanting.

If you use alchemy and vampirism/necromage, you can achieve 33%, I won't go into that here though. It takes 6 enchanting perks, 7 alchemy perks (I think, might be 6), and 3 restoration perks.