We　investigate　the　stability　and　instability　of　the　magnetic　Rayleigh–Bénard　problem　with　zero　resistivity.　A　stability　criterion　is　established,　under　which　the　magnetic　Bénard　problem　is　stable.　The　proof　is　mainly　based　on　a　three-layers　energy　method　and　an　idea　of　magnetic　inhibition　mechanism.　The　stability　result　first　mathematically　verifies　Chandrasekhar＇s　physical　conjecture　in　1955　that　the　thermal　instability　can　be　inhibited　by　a　strong　(impressed)　magnetic　field　in　magnetohydrodynamic　(MHD)　fluids　with　zero　resistivity　(based　on　a　linearized　steady　magnetic　Bénard　equations).　In　addition,　we　also　provide　an　instability　criterion,　under　which　the　magnetic　Rayleigh–Bénard　problem　is　unstable.　The　instability　proof　is　mainly　based　on　a　bootstrap　instability　method　by　further　developing　new　techniques.　Our　instability　result　shows　that　the　thermal　instability　still　occurs　when　the　(impressed)　magnetic　field　is　weak.　©　2020　Elsevier　Masson　SAS