Frost mages freeze enemies in their tracks and shatter them with icy magic, aided by summoned elementals or conjured wards.
When you choose this specialization at 2nd level, you gain the Shape Water cantrip, which doesn't count against the number of mage cantrips you know.
Also at 2nd level, you learn to manifest your icy magic in an especially potent form. Choose one of the following features.
When you cast an spell of 1st level or higher that deals frost damage, you can simultaneously use a strand of the spell's magic to create an ice barrier on yourself that lasts until you finish a long rest. The barrier has hit points equal to twice your mage level + your Intelligence modifier. Whenever you take damage, the barrier takes the damage instead. If this damage reduces the barrier to 0 hit points, you take any remaining damage.
While the ice barrier has 0 hit points, it can't absorb damage, but its magic remains. Whenever you cast spell of 1st level or higher that deals cold damage, the barrier regains a number of hit points equal to twice the level of the spell.
Once you create the ice barrier, you can't create it again until you finish a long rest.
You can use your action and spend a spell slot of 1st level or higher to summon a water elemental to your service. The statblock for such is currently listed here, and it uses your proficiency bonus (PB) in several places.
In combat, the summoned water elemental acts during your turn. It can move and use its reaction on its own, but the only action it takes is the Dodge action, unless you take a bonus action on your turn to command it to take another action. That action can be one in its stat block or some other action. If you are incapacitated, the elemental can take any action of its choice, not just Dodge.
If your elemental drops to 0 hit points or you dismiss it as an action, its form disperses. You can only have one summoned water elemental at a time; summoning a new one causes an existing elemental to immediately disperse.
Beginning at 6th level, you can freeze your enemies with the icy magic of your spells. Whenever you deal cold damage to a creature with a spell attack, or a creature fails the saving throw of a spell you cast that deals cold damage, that target is chilled until the end of your next turn: its speed is halved, and it can't take reactions. Only your spells of 1st level or higher can chill creatures.
In addition, whenever a creature you have chilled takes cold damage from one of your spells, add your Intelligence modifier to the damage (minimum 1).
When you reach 10th level, your signature manifestation of frost magic become even more potent. You gain one of the following features, depending on your choice at 2nd level.
You can harness the lingering icy magic of your spells to unleash on your enemies. Whenever you cast spell of 1st level or higher that deals cold damage, instead of restoring hit points to your ice barrier, you can use your bonus action to make a spell attack roll against a creature you can see within 60 feet. On a hit, the creature takes cold damage equal to 1d6 + twice the level of the spell.
Your summoned water elemental gains the following benefits:
Any creature hit by its Slam attack is grappled, with an escape DC equal to your spell save DC. The elemental can only grapple one creature at a time.
Creatures hit by its waterbolt are pushed up to 10 feet away from the elemental in a straight line.
When it uses its Freeze action, creatures that fail their saving throw take cold damage equal to 2d6 + your proficiency bonus, and are restrained until the end of the elemental's next turn. Creatures that succeed on their save take half damage and suffer no other effects.
At 14th level, your magical attacks against frozen foes become far deadlier. Your spell attack rolls have advantage against creatures you have chilled, and such creatures have disadvantage on saving throws against your spells.