Ice Troll (RotFM) Cold Aura vs. Wild Shape Fire Elemental

In Rime of the Frost Maiden, the Ice Troll has the following ability:

While it’s alive, the troll generates an aura of bitter cold that fills the area within 10 feet of it. At the start of the troll’s turn, all nonmagical flames in the aura are extinguished. Any creature that starts its turn within 10 feet of the troll takes 10 (3d6) cold damage.

Meanwhile I have a Druid who can Wildshape into a Fire Elemental. Would that be considered magical fire? Also, if the Troll is ignited due to the Fire Elemental, would that extinguish the flames?