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Alteration
Changes the physical world. Alteration spells can give units increased strength,
resistance to fire, and the like. They can also change the weather. - Illwinter.
Alteration is a very handy school. It's best-known for its "buff" spells, which alter your teammates to make them significantly tougher to harm through conventional means. Barkskin, Stoneskin, and Ironskin are the trademark examples, but there are other options provided by other paths. There are also "debuff" spells, such as Blindness & Shrink, but not as many.
This school also provides a fair amount of attack spells, altering targets in ways that deal severe damage to them (such as Combustion and Incinerate). Deep in the school, you'll find methods of converting foes into useless forms, such as a pathetic animal (Curse of the Frog Prince, Polymorph) or a pile (Bone Melter, Liquify, Disintegrate).
Rituals also change the form of things; Pyromancers & Geomancers can use their Gems to transmute stuff into Gold, Nature Mages can change their own forms in an attempt to learn a different perspective of magic, and super-powerful Astral Mages will eventually be able to derive whatever they want from the universe (see Wish for more details). Earth Mages can also alter Fortress walls if they're skilled enough (Crumble, Iron Walls), and will even be able to raise forts in a single turn (Wizard's Tower), though these Rituals are deep in the school.
Generalization
Standard
Fire primarily allows you to set fire to stuff (even allowing you to prime yourself to combust in a pinch), but it also provides
Cold Resistance.
Air can shape the wind to help you out, but is best known for turning teammates into fog or creating
Phantasmal Warrior illusions out of thin air.
Water primarily allows you to freeze/slow and melt/quicken stuff (the former is more effective underwater), but it also provides
Fire Resistance.
Earth is the source of
Stoneskin and
Ironskin, along with strengthening magic, but it can also be used to catch others with the ground or to destroy gear. It also provides
Shock Resistance, and can provide other resistances on a personal basis.
Astral primarily allows you to alter your fate (
Twist Fate,
Luck), but can also be used to alter time (
Time Stop) and space (
Body Ethereal). It also allows you to alter the allegiances of
Magic Beings.
Death lets you alter the life force of beings, making them weaker (
Weakness) or even stealing their HP (
Drain Life), but it also lets you make things more "dead" (
Hand of Dust,
Disintegrate, and technically
Skeletal Legion). The caster can use it to make themselves
Invulnerable and/or
Pierce-Resistant.
Nature is the source of
Barkskin, but it also allows you to alter beings in a less-drastic manner. You can make them bigger (
Enlarge,
Swarm,
Mother Oak), smaller (
Shrink), or even entirely-different beings. It also allows the caster to self-apply a broad array of Elemental Resistances, such as
Poison Resistance.
Crosspaths
Combat Magic
Rituals