diff --git a/core/src/air/extension.rs b/core/src/air/extension.rs
index 1a546c8b16..6e8401bbdc 100644
--- a/core/src/air/extension.rs
+++ b/core/src/air/extension.rs
@@ -19,9 +19,9 @@ pub const DEGREE: usize = 4;
 #[repr(C)]
 pub struct Extension<T>(pub [T; DEGREE]); // Degree 4 is hard coded for now.  TODO:  Change to a const generic
 
-impl<V> Extension<V> {
+impl<E: AbstractField> Extension<E> {
     // Returns the one element of the extension field
-    pub fn one<AB: SP1AirBuilder<Var = V>>() -> Extension<AB::Expr>
+    pub fn one<AB: SP1AirBuilder<Expr = E>>() -> Extension<AB::Expr>
     where
         AB::Expr: AbstractField,
     {
@@ -30,51 +30,72 @@ impl<V> Extension<V> {
     }
 
     // Converts a field element to extension element
-    pub fn from<AB: SP1AirBuilder<Var = V>>(x: V) -> Extension<AB::Expr>
-    where
-        AB::Expr: From<V>,
-    {
-        Extension(field_to_array(x.into()))
+    pub fn from<AB: SP1AirBuilder<Expr = E>>(x: E) -> Extension<AB::Expr> {
+        Extension(field_to_array(x))
     }
 
     // Negates an extension field Element
-    pub fn neg<AB: SP1AirBuilder<Var = V>>(self) -> Extension<AB::Expr> {
+    pub fn neg<AB: SP1AirBuilder<Expr = E>>(self) -> Extension<AB::Expr> {
         Extension(self.0.map(|x| AB::Expr::zero() - x))
     }
 
     // Adds an extension field element
-    pub fn add<AB: SP1AirBuilder<Var = V>>(self, rhs: &Self) -> Extension<AB::Expr>
+    pub fn add<AB: SP1AirBuilder<Expr = E>>(self, rhs: &Self) -> Extension<AB::Expr>
     where
-        V: Add<V, Output = AB::Expr> + Copy,
+        E: Add<E, Output = AB::Expr>,
     {
         let mut elements = Vec::new();
 
-        for (e1, e2) in self.0.into_iter().zip_eq(rhs.0.into_iter()) {
+        for (e1, e2) in self.0.into_iter().zip_eq(rhs.0.clone().into_iter()) {
             elements.push(e1 + e2);
         }
 
         Extension(elements.try_into().unwrap())
     }
 
+    // Subtracts an extension field element
+    pub fn sub<AB: SP1AirBuilder<Expr = E>>(self, rhs: &Self) -> Extension<AB::Expr>
+    where
+        E: Add<E, Output = AB::Expr>,
+    {
+        let mut elements = Vec::new();
+
+        for (e1, e2) in self.0.into_iter().zip_eq(rhs.0.clone().into_iter()) {
+            elements.push(e1 - e2);
+        }
+
+        Extension(elements.try_into().unwrap())
+    }
+
     // Multiplies an extension field element
-    pub fn mul<AB: SP1AirBuilder<Var = V>>(self, rhs: &Self) -> Extension<AB::Expr>
+    pub fn mul<AB: SP1AirBuilder<Expr = E>>(self, rhs: &Self) -> Extension<AB::Expr>
     where
-        V: Mul<V, Output = AB::Expr> + Copy,
+        E: Mul<E, Output = AB::Expr>,
     {
         let mut elements = Vec::new();
 
-        for (e1, e2) in self.0.into_iter().zip_eq(rhs.0.into_iter()) {
+        for (e1, e2) in self.0.into_iter().zip_eq(rhs.0.clone().into_iter()) {
             elements.push(e1 * e2);
         }
 
         Extension(elements.try_into().unwrap())
     }
 
-    pub fn as_base_slice(&self) -> &[V] {
+    pub fn as_base_slice(&self) -> &[E] {
         &self.0
     }
 }
 
+impl<V> Extension<V> {
+    // Converts a field element with var base elements to one with expr base elements.
+    pub fn from_var<AB: SP1AirBuilder<Var = V>>(self) -> Extension<AB::Expr>
+    where
+        V: Into<AB::Expr>,
+    {
+        Extension(self.0.map(|x| x.into()))
+    }
+}
+
 impl<F> From<BinomialExtensionField<F, 4>> for Extension<F>
 where
     F: Field,
diff --git a/core/src/operations/div_extension.rs b/core/src/operations/div_extension.rs
index a05fd65650..40abaeb340 100644
--- a/core/src/operations/div_extension.rs
+++ b/core/src/operations/div_extension.rs
@@ -2,11 +2,8 @@
 //!
 use core::borrow::Borrow;
 use core::borrow::BorrowMut;
-use p3_air::AirBuilder;
 use p3_field::extension::BinomialExtensionField;
 use p3_field::extension::BinomiallyExtendable;
-use p3_field::AbstractField;
-use p3_field::Field;
 use sp1_derive::AlignedBorrow;
 use std::mem::size_of;
 
@@ -14,10 +11,14 @@ use crate::air::Extension;
 use crate::air::SP1AirBuilder;
 use crate::air::DEGREE;
 
+use super::IsEqualExtOperation;
+
 /// A set of columns needed to compute whether the given word is 0.
 #[derive(AlignedBorrow, Default, Debug, Clone, Copy)]
 #[repr(C)]
 pub struct DivExtOperation<T> {
+    pub is_equal: IsEqualExtOperation<T>,
+
     /// Result is the quotient
     pub result: Extension<T>,
 }
@@ -30,6 +31,10 @@ impl<F: BinomiallyExtendable<DEGREE>> DivExtOperation<F> {
     ) -> BinomialExtensionField<F, DEGREE> {
         let result = a / b;
         self.result = result.into();
+
+        let product = b * result;
+        self.is_equal.populate(a, product);
+
         result
     }
 
@@ -39,16 +44,12 @@ impl<F: BinomiallyExtendable<DEGREE>> DivExtOperation<F> {
         b: Extension<AB::Expr>,
         cols: DivExtOperation<AB::Var>,
         is_real: AB::Expr,
-    ) {
+    ) where
+        AB::F: BinomiallyExtendable<DEGREE>,
+    {
         builder.assert_bool(is_real.clone());
 
-        let product = b.mul(&cols.result);
-        builder.when(is_real.clone()).assert_eq(product, a);
-
-        // If the result is 1, then the input is 0.
-        builder
-            .when(is_real.clone())
-            .when(cols.result)
-            .assert_zero(a.clone());
+        let product = b.mul::<AB>(&cols.result.from_var::<AB>());
+        IsEqualExtOperation::<AB::F>::eval(builder, a, product, cols.is_equal, is_real.clone());
     }
 }
diff --git a/core/src/operations/is_equal_extension.rs b/core/src/operations/is_equal_extension.rs
index 9f95f59de3..bd2361959c 100644
--- a/core/src/operations/is_equal_extension.rs
+++ b/core/src/operations/is_equal_extension.rs
@@ -36,11 +36,13 @@ impl<F: BinomiallyExtendable<DEGREE>> IsEqualExtOperation<F> {
         b: Extension<AB::Expr>,
         cols: IsEqualExtOperation<AB::Var>,
         is_real: AB::Expr,
-    ) {
+    ) where
+        AB::F: BinomiallyExtendable<DEGREE>,
+    {
         builder.assert_bool(is_real.clone());
 
         // Calculate differences.
-        let diff = a.sub(b);
+        let diff = a.sub::<AB>(&b);
 
         // Check if the difference is 0.
         IsZeroExtOperation::<AB::F>::eval(builder, diff, cols.is_diff_zero, is_real.clone());