Reading the PHB chapter 5 equipment pg. 151. It describes an arcane focus as such:
Arcane Focus. An arcane focus is a special item — an orb, a crystal, a rod, a specially constructed staff, a wand like length of wood, or some similar item — designed to channel the power of arcane spells. A sorcerer, warlock, or wizard can use such an item as a spellcasting focus, as described in chapter 10.
Now I have been coming up with a dwarven war mage character concept, and was curious to know would it be allowed to have an arcane focus as warhammer?
Arcane Focus is a Specific Item
The arcane focus is found in the Adventuring Gear section of the equipment chapter (PH page 150, SRD page 69, SRD link). Your options for arcane focus are:
Crystal
Orb
Rod
Staff
Wand
So no, you cannot use a warhammer as a focus.
Why Not1?
An arcane focus is “…an orb, a crystal, a rod, a specially constructed staff, a wand-like length of wood, or some similar item….” (emphasis mine). A warhammer or other weapon is not similar to any of the listed items.
Can A DM allow it? You Bet
That being said, if your DM says it’s OK, go for it. Paladins and clerics can use a holy symbol as a focus, and that can be “…carefully engraved or inlaid as an emblem on a shield….” Using that as precedence, you can thusly allow arcane magic users to use the same rules.
Just know that in so doing, you are giving the arcane magic users an ability they didn’t have before and devaluing the divine caster’s unique schtik.
1 An exception that requires a common magic item
If the warhammer has a Ruby of the War Mage imbedded in it. The Ruby of the War Mage (Xanathar’s Guide to Everything) is a magical gem that can be attached to a weapon, at which point that weapon can be used as a spellcasting focus. Things to note
it says “spellcasting focus”, not “arcane” or “divine”, implying that it works with all spellcasting in the same way as a comnponents pouch
it’s listed as a common item, implying that it’s not considered too much of a big deal
it requires attunement