In this paper we have developed the quantum theory of a relativistic electron gas in magnetic fields of arbitrary strength, and we have obtained the equation of state for such a gas. Intense magnetic fields (of field strengths ^10 13 G) are believed to exist during gravitational collapse and in collapsed astronomical objects. Our results show that the stress tensor is very anisotropic when (p z 2 )/m 2 c 2 \_p z -(z momentum) = (momentum parallel to the field)] is smaller than (2H/H^) l t % t where H is the field and E c -m 2 c % l'he-^A\^ X10 13 G. At a temperature of 10 9 °K or a density of 10 6 g/cm 3 , the theory developed here must be used if #>10 12 G.

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