I Play a 2nd level Wizard/ 3rd level fighter (Eldrich Knight). This makes me a 3rd lvl caster, with 2ndlevel spell slots. Yet I can't add 2nd level spells to my spellbook, even though that should be possible according to this rule:
Learning Spells of 1st Level and Higher
Each time you gain a wizard level, you can add two wizard spells of your choice to your spellbook for free. Each of these spells must be of a level for which you have spell slots, as shown on the Wizard table. On your adventures, you might find other spells that you can add to your spellbook (see the “Your Spellbook” sidebar).
Each time you gain a wizard level, you can add two wizard spells of your choice to your spellbook for free. Each of these spells must be of a level for which you have spell slots, as shown on the Wizard table. On your adventures, you might find other spells that you can add to your spellbook (see the “Your Spellbook” sidebar).
You have the spellslots of a multiclass caster, but you learn spells individually for each class. So you can learn the spells of a Level 2 Wizard or a Level 3 Eldritch Knight, not of a Level 3 full caster.
You can use those 2nd level spell slots to upcast your 1st level spells.
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I Play a 2nd level Wizard/ 3rd level fighter (Eldrich Knight). This makes me a 3rd lvl caster, with 2ndlevel spell slots.
Yet I can't add 2nd level spells to my spellbook, even though that should be possible according to this rule:
No, you can known spells according to your classes taken singularly. This is explained in Multiclass.
You have the spellslots of a multiclass caster, but you learn spells individually for each class. So you can learn the spells of a Level 2 Wizard or a Level 3 Eldritch Knight, not of a Level 3 full caster.
You can use those 2nd level spell slots to upcast your 1st level spells.