For some wizards, the study of magic is just one study. An area of emphasis to be sure, but just one of many. These wizards sacrifice the magical specialization of their peers in order to pursue a well-rounded body of knowledge. Their erstwhile peers may call them dilettantes, but the Arcane Scholar knows better than anyone that knowledge is power.

Polymath

At 2nd level, you've already started broadening your studies. You gain proficiency in two skills of your choice and a language.

Additionally, you've been able to learn a lot about a certain subject, such as animals or psychology, allowing you to augment your intuition with knowledge. Choose Survival, Insight, Medicine, or Animal Handling. You may add your intelligence to your skill check with the chosen skill.

Animal Handling

Add your Intelligence modifier to your Animal Handling checks.

Insight

Add your Intelligence modifier to your Insight checks.

Medicine

Add your Intelligence modifier to your Medicine checks.

Survival

Add your Intelligence modifier to your Survival checks.

Half the Battle

You've studied deeply into the lore of your potential enemies. When you roll initiative, you may select one enemy you can see that you haven't used this skill on since your last long rest. Make a check with a skill based on the target's creature type. The DC is equal to 10 + half their CR or level.

Arcana: Aberrations, Constructs, Undead, Elemental, Fey

History: Humanoids, Giants

Nature: Beasts, Dragon, Monstrosity, Ooze, Plant

Religion: Celestial, Fiend

You gain knowledge about 1 resistance, immunity, vulnerability, or ability that would give them advantage or disadvantage to a status or saving throw, plus one more for every 1 you beat the check by. If you roll a 20, you learn all of them, and may select another enemy to make the roll against as well.

This check automatically fails against enemies that have disguised their appearance or have never been seen before, and cannot discover anything temporary, such as a resistance from a spell or potion.

Sharp Mind

Constant study has sharpened your mind to a razor edge. You have resistance against Psychic damage, as well as advantage on Charisma, Wisdom, and Intelligence saving throws against effects that would inflict a madness; reduce your intelligence, wisdom, or charisma; or have a psionic origin.

Cause and Effect

At 6th level, you have studied details of your spells, and know better than anyone else what they can or can't do. When you ask the DM a question about what would happen if you cast a wizard spell you know, they must answer truthfully according to knowledge you would reasonably have. This information cannot be how a being would react or effects that are inherently unpredictable (such as what element a Chaos Bolt would be), and may not be correct if there is a factor you do not know about. For example, if you ask if casting Enlarge/Reduce on a giant to would them collapse a bridge under their weight, they may tell you no when it actually would, because of a structural flaw you have no way of knowing about.

Master Sage

At 14th level, there are few in the world more knowledgeable than you. When you fail a knowledge check in a skill your are proficient with, or any check with the skill you chose to add your intelligence modifier to at level 2, you may reroll it. You may only do this once per check, cannot use it in the same check as your use Lucky, and may only do a number of times per long rest equal to your proficiency bonus.