A brick pizza oven dome is usually constructed as either an igloo or barrel dome. Both shapes are constructed in much the same fashion with bricks, mortar and usually with a supporting frame. Construction of an igloo dome does not necessarily require a frame if completed by a competent brick layer. Fireclay or refractory mortar is not normally required. Some cracks may develop with regular building mortar however these will be within the oven and not readily noticeable.
After constructing the internal brick wall you will need to cover the brick with an insulation layer to further improve the heat retaining properties of the oven.