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At1st level Eldritch Knight obtains the following ability:

Diverse Training: An eldritch knight adds his level to any levels of fighter he might have for the purpose of meeting the prerequisites for feats (if he has no fighter levels, treat his eldritch knight levels as levels of fighter). He also adds his level to any levels in an arcane spellcasting class for the purpose of meeting the prerequisites for feats.

While his Spells Per Day ability specifically explains what to do in case of multiclassing with more than one Arcane spellcasting class, this ability does not.

Does the Eldritch Knight add his EK level to any arcane spellcasting class he also has for the purpose of qualifying for feats? If not and only a single class gets that bonus, can that class be different from the one benefitting from Spells Per Day? Finally, can he add some of his level to one and some to another?

Considering an example: a Magus 3/Wizard 10/EK 2. Is it possible for that character to take the Extra Arcana feat and select Broad Study (Wizard) while advancing to the 3rd level in Eldritch Knight?

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No, for prestige classes whenever they advance an attribute it is solely the attribute of the class used to meet the prerequisites. In this case the "arcane spell casting class" would be the wizard, because the wizard was (presumably) the source of your third level spells.

To clarify it further, it notes that "he adds his levels to any levels in AN arcane spell casting class" - emphasis mine. Diverse training only affects one class, not multiple classes.

The disclosure under the Spells per Day is a bit redundant, as there are no prestige classes (that I can think of) which stack abilities with multiple classes simultaneously, especially classes that weren't used to qualify for the prestige class in the first place (I discount of course BAB and saves).

Best of luck in your eldritch pursuits, unfortunately this path is not open (RAW).

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Short version: no, but yes for extra arcana.

Long version:

It you were able to add your EK level to all your arcane spellcasting classes, it would have be worded explicitly as "adds his level to any levels in all arcane spellcasting class he possesses", but that is not the case. You have to choose one spellcasting class and stick to it. RAW it technically don't have to be the same that the one benefitting from Spells per Day, but your GM will probably patch that.

But...

The good news is that you are not completely screwed. You can totally use your lvl3 EK feat to take extra arcana even if you oriented EK to wizard, as the prerequisite is a class feature you got. However Broad Study is a hard requirement of 6 levels of Magus, and that trick won't do it.

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