Cryomancer

Many disciplines of the arcane teach their students how to add energy into the world. Cryomancy, however, teaches the art of pulling energy away. Specifically, the school of magic teaches how to siphon away thermal energies. When a wizard can simply freeze a threat in place, it no longer becomes a threat. Many foolishly only see this magic as beautiful, and it is, but within the beauty of all that shimmering ice, is a slow and shivering death.

Frigid Magics

When you choose this school at 3rd level, you learn to pull the thermal energy out of the air with your spells, chilling your enemies to the bone. Whenever you deal cold damage, you deal additional damage equal to your Proficiency bonus.

When you cast a Wizard spell of 1st level or higher, you can temporarily replace its damage type with cold damage as a free action for that casting. You may replace the damage type a number of times equal to your Intelligence modifier per short rest.

Finally, you gain the ray of frost cantrip if you don't already know it, and it is thereafter always prepared.

Frozen Solid

Starting at 3rd level, your icy magic slows the world down to a crawl. Whenever you would deal cold damage to a creature with a spell of 1st level or higher, you may prevent the damage and instead have the creature make a Constitution saving throw against your spell casting DC. If they fail their save, they become Frozen.

A creature that is Frozen is restrained and takes 1d4 additional damage from bludgeoning, force, and cold damage sources.

At the end of each of their turns, a Frozen creature makes a Strength save against your spell casting DC to attempt to break free of the ice. If they succeed, they are no longer Frozen, otherwise they take 1d4 cold damage.

Chilled Winds

Starting at 6th level, painful cold winds swirl around you and bite into any would-be assailant. You now produce a 5 foot Emanation of icy cold. Whenever you cast a spell of 1st level or higher, this Emanation's size doubles to 10 feet.

Whenever the Emanation enters a creature’s space and whenever a creature enters the Emanation or ends its turn there, you may either have the creature take cold damage equal to your Intelligence modifier or have them make a Constitution saving throw against your spell casting DC. If they fail their save, they become Frozen.

Cold Snap

Beginning at 10th level, cold magic runs through you with such fervor that it bursts forth from your icy victims. Frozen creatures now produce a 10 foot Emanation of frigid cold.

Whenever the Emanation enters a creature’s space and whenever a creature enters the Emanation or ends its turn there, you may either have the creature take cold damage equal to your Intelligence modifier or have them make a Constitution saving throw against your spell casting DC. If they fail their save, they become Frozen.

Absolute Zero

By 14th level, your cold magic surges through your body, lowering the surrounding temperatures to impossibly cold levels. You leech the warmth out of everything around you, plants frost over and die at your touch, and snow falls around you and wherever you walk.

Whenever a creature makes a Constitution saving throw against becoming Frozen, or makes a Strength save to end being Frozen, you may have that creature make their save with disadvantage. You may use this feature a number of times equal to your Proficiency Bonus per long rest. 

A creature that is Frozen now takes 1d8 additional damage from bludgeoning, force, and cold damage sources, instead of 1d4; A creature failing their Strength save to end being Frozen now takes 1d8 cold damage instead of 1d4.