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We use our recently proposed algebraic approach for calculating the heatkernel associated with the Laplace operator to calculate the one-loop effectiveaction in the non-Abelian gauge theory. We consider the most general case ofarbitrary space-time dimension, arbitrary compact simple gauge group andarbitrary matter and assume a covariantly constant gauge field strength of themost general form, having many independent color and space-time invariants(Savvidy type chromomagnetic vacuum) and covariantly constant scalar fields asa background. The explicit formulas for all the needed heat kernels andzeta-functions are obtained. We propose a new method to study the vacuumstability and show that the background field configurations with covariantlyconstant chromomagnetic fields can be stable only in the case when more thanone independent field invariants are present and the values of these invariantsdiffer not greatly from each other. The role of space-time dimension isanalyzed in this connection and it is shown that this is possible only inspace-times with dimensions not less than five $d\geq 5$.Comment: 14 pages, LATeX, University of Greifswald (1994

Covariant algebraic calculation of the one-loop effective potential in non-Abelian gauge theory and a new approach to stability problem