diff --git a/src/core/backend/avx512/cm31.rs b/src/core/backend/avx512/cm31.rs
index e76fbddbb..b2a060a65 100644
--- a/src/core/backend/avx512/cm31.rs
+++ b/src/core/backend/avx512/cm31.rs
@@ -1,9 +1,9 @@
-use std::ops::{Add, Mul, MulAssign, Sub};
+use std::ops::{Add, Mul, MulAssign, Neg, Sub};
 
 use num_traits::{One, Zero};
 
 use super::m31::{PackedBaseField, K_BLOCK_SIZE};
-use crate::core::fields::cm31::{CM31, P2};
+use crate::core::fields::cm31::CM31;
 use crate::core::fields::FieldExpOps;
 
 /// AVX implementation for the complex extension field of M31.
@@ -69,10 +69,17 @@ impl MulAssign for PackedCM31 {
         *self = *self * rhs;
     }
 }
+impl Neg for PackedCM31 {
+    type Output = Self;
+    fn neg(self) -> Self::Output {
+        Self([-self.a(), -self.b()])
+    }
+}
 impl FieldExpOps for PackedCM31 {
     fn inverse(&self) -> Self {
         assert!(!self.is_zero(), "0 has no inverse");
-        self.pow((P2 - 2) as u128)
+        // 1 / (a + bi) = (a - bi) / (a^2 + b^2).
+        Self([self.a(), -self.b()]) * (self.a().square() + self.b().square()).inverse()
     }
 }
 
diff --git a/src/core/backend/avx512/m31.rs b/src/core/backend/avx512/m31.rs
index af07610a6..fa965c4f9 100644
--- a/src/core/backend/avx512/m31.rs
+++ b/src/core/backend/avx512/m31.rs
@@ -254,6 +254,8 @@ impl One for PackedBaseField {
 
 impl FieldExpOps for PackedBaseField {
     fn inverse(&self) -> Self {
+        // TODO(andrew): Use a better multiplication tree. Also for other constant powers in the
+        // code.
         assert!(!self.is_zero(), "0 has no inverse");
         self.pow((P - 2) as u128)
     }
diff --git a/src/core/backend/avx512/qm31.rs b/src/core/backend/avx512/qm31.rs
index e0d06ffd4..f63d8df61 100644
--- a/src/core/backend/avx512/qm31.rs
+++ b/src/core/backend/avx512/qm31.rs
@@ -6,7 +6,7 @@ use num_traits::{One, Zero};
 use super::cm31::PackedCM31;
 use super::m31::K_BLOCK_SIZE;
 use super::PackedBaseField;
-use crate::core::fields::qm31::{P4, QM31};
+use crate::core::fields::qm31::QM31;
 use crate::core::fields::FieldExpOps;
 
 /// AVX implementation for an extension of CM31.
@@ -125,10 +125,13 @@ impl MulAssign for PackedSecureField {
 }
 impl FieldExpOps for PackedSecureField {
     fn inverse(&self) -> Self {
-        // TODO(andrew): Use a better multiplication tree. Also for other constant powers in the
-        // code.
         assert!(!self.is_zero(), "0 has no inverse");
-        self.pow(P4 - 2)
+        // (a + bu)^-1 = (a - bu) / (a^2 - (2+i)b^2).
+        let b2 = self.b().square();
+        let ib2 = PackedCM31([-b2.b(), b2.a()]);
+        let denom = self.a().square() - (b2 + b2 + ib2);
+        let denom_inverse = denom.inverse();
+        Self([self.a() * denom_inverse, -self.b() * denom_inverse])
     }
 }
 
diff --git a/src/core/backend/cpu/mod.rs b/src/core/backend/cpu/mod.rs
index cee638a38..5edcfc469 100644
--- a/src/core/backend/cpu/mod.rs
+++ b/src/core/backend/cpu/mod.rs
@@ -57,6 +57,7 @@ mod tests {
     use rand::rngs::SmallRng;
 
     use crate::core::backend::{CPUBackend, Column, FieldOps};
+    use crate::core::fields::m31::P;
     use crate::core::fields::qm31::QM31;
     use crate::core::fields::FieldExpOps;
 
@@ -66,10 +67,10 @@ mod tests {
         let column: Vec<QM31> = (0..16)
             .map(|_| {
                 QM31::from_u32_unchecked(
-                    rng.gen::<u32>(),
-                    rng.gen::<u32>(),
-                    rng.gen::<u32>(),
-                    rng.gen::<u32>(),
+                    rng.gen::<u32>() % P,
+                    rng.gen::<u32>() % P,
+                    rng.gen::<u32>() % P,
+                    rng.gen::<u32>() % P,
                 )
             })
             .collect();
@@ -88,10 +89,10 @@ mod tests {
         let column: Vec<QM31> = (0..16)
             .map(|_| {
                 QM31::from_u32_unchecked(
-                    rng.gen::<u32>(),
-                    rng.gen::<u32>(),
-                    rng.gen::<u32>(),
-                    rng.gen::<u32>(),
+                    rng.gen::<u32>() % P,
+                    rng.gen::<u32>() % P,
+                    rng.gen::<u32>() % P,
+                    rng.gen::<u32>() % P,
                 )
             })
             .collect();
diff --git a/src/core/fields/cm31.rs b/src/core/fields/cm31.rs
index c585855b6..b1f8e11fc 100644
--- a/src/core/fields/cm31.rs
+++ b/src/core/fields/cm31.rs
@@ -52,6 +52,14 @@ impl Mul for CM31 {
     }
 }
 
+impl FieldExpOps for CM31 {
+    fn inverse(&self) -> Self {
+        assert!(!self.is_zero(), "0 has no inverse");
+        // 1 / (a + bi) = (a - bi) / (a^2 + b^2).
+        Self(self.0, -self.1) * (self.0.square() + self.1.square()).inverse()
+    }
+}
+
 #[cfg(test)]
 #[macro_export]
 macro_rules! cm31 {
@@ -66,9 +74,16 @@ mod tests {
 
     use super::CM31;
     use crate::core::fields::m31::P;
-    use crate::core::fields::IntoSlice;
+    use crate::core::fields::{FieldExpOps, IntoSlice};
     use crate::m31;
 
+    #[test]
+    fn test_inverse() {
+        let cm = cm31!(1, 2);
+        let cm_inv = cm.inverse();
+        assert_eq!(cm * cm_inv, cm31!(1, 0));
+    }
+
     #[test]
     fn test_ops() {
         let cm0 = cm31!(1, 2);
diff --git a/src/core/fields/m31.rs b/src/core/fields/m31.rs
index 20fb92424..37809cbc4 100644
--- a/src/core/fields/m31.rs
+++ b/src/core/fields/m31.rs
@@ -78,6 +78,13 @@ impl Mul for M31 {
     }
 }
 
+impl FieldExpOps for M31 {
+    fn inverse(&self) -> Self {
+        assert!(!self.is_zero(), "0 has no inverse");
+        self.pow(P as u128 - 2)
+    }
+}
+
 impl ComplexConjugate for M31 {
     fn complex_conjugate(&self) -> Self {
         *self
diff --git a/src/core/fields/mod.rs b/src/core/fields/mod.rs
index 2ab0c9c37..b35d49f80 100644
--- a/src/core/fields/mod.rs
+++ b/src/core/fields/mod.rs
@@ -231,13 +231,6 @@ macro_rules! impl_field {
             }
         }
 
-        impl FieldExpOps for $field_name {
-            fn inverse(&self) -> Self {
-                assert!(!self.is_zero(), "0 has no inverse");
-                self.pow(($field_size - 2) as u128)
-            }
-        }
-
         impl Product for $field_name {
             fn product<I>(mut iter: I) -> Self
             where
diff --git a/src/core/fields/qm31.rs b/src/core/fields/qm31.rs
index 3862ba321..9b739b15f 100644
--- a/src/core/fields/qm31.rs
+++ b/src/core/fields/qm31.rs
@@ -66,6 +66,18 @@ impl Mul for QM31 {
     }
 }
 
+impl FieldExpOps for QM31 {
+    fn inverse(&self) -> Self {
+        assert!(!self.is_zero(), "0 has no inverse");
+        // (a + bu)^-1 = (a - bu) / (a^2 - (2+i)b^2).
+        let b2 = self.1.square();
+        let ib2 = CM31(-b2.1, b2.0);
+        let denom = self.0.square() - (b2 + b2 + ib2);
+        let denom_inverse = denom.inverse();
+        Self(self.0 * denom_inverse, -self.1 * denom_inverse)
+    }
+}
+
 #[cfg(test)]
 #[macro_export]
 macro_rules! qm31 {
@@ -77,13 +89,21 @@ macro_rules! qm31 {
 
 #[cfg(test)]
 mod tests {
+    use num_traits::One;
     use rand::Rng;
 
     use super::QM31;
     use crate::core::fields::m31::P;
-    use crate::core::fields::IntoSlice;
+    use crate::core::fields::{FieldExpOps, IntoSlice};
     use crate::m31;
 
+    #[test]
+    fn test_inverse() {
+        let qm = qm31!(1, 2, 3, 4);
+        let qm_inv = qm.inverse();
+        assert_eq!(qm * qm_inv, QM31::one());
+    }
+
     #[test]
     fn test_ops() {
         let qm0 = qm31!(1, 2, 3, 4);