diff --git a/src/field/extension_field/algebra.rs b/src/field/extension_field/algebra.rs
new file mode 100644
index 00000000..db80ce75
--- /dev/null
+++ b/src/field/extension_field/algebra.rs
@@ -0,0 +1,153 @@
+use crate::field::extension_field::{FieldExtension, OEF};
+use std::fmt::{Debug, Display, Formatter};
+use std::iter::{Product, Sum};
+use std::ops::{Add, AddAssign, Mul, MulAssign, Neg, Sub, SubAssign};
+
+/// Let `F_D` be the extension `F[X]/(X^D-W)`. Then `ExtensionAlgebra<F_D>` is the quotient `F_D[X]/(X^D-W)`.
+/// It's a `D`-dimensional algebra over `F_D` useful to lift the multiplication over `F_D` to a multiplication over `(F_D)^D`.
+#[derive(Copy, Clone)]
+pub struct ExtensionAlgebra<F: OEF<D>, const D: usize>([F; D]);
+
+impl<F: OEF<D>, const D: usize> ExtensionAlgebra<F, D> {
+    pub const ZERO: Self = Self([F::ZERO; D]);
+
+    pub fn one() -> Self {
+        F::ONE.into()
+    }
+
+    pub fn from_basefield_array(arr: [F; D]) -> Self {
+        Self(arr)
+    }
+
+    pub fn to_basefield_array(self) -> [F; D] {
+        self.0
+    }
+}
+
+impl<F: OEF<D>, const D: usize> From<F> for ExtensionAlgebra<F, D> {
+    fn from(x: F) -> Self {
+        let mut arr = [F::ZERO; D];
+        arr[0] = x;
+        Self(arr)
+    }
+}
+
+impl<F: OEF<D>, const D: usize> Display for ExtensionAlgebra<F, D> {
+    fn fmt(&self, f: &mut Formatter<'_>) -> std::fmt::Result {
+        write!(f, "({}) + ", self.0[0])?;
+        for i in 1..D - 1 {
+            write!(f, "({})*b^{} + ", self.0[i], i)?;
+        }
+        write!(f, "{}*b^{}", self.0[D - 1], D - 1)
+    }
+}
+
+impl<F: OEF<D>, const D: usize> Debug for ExtensionAlgebra<F, D> {
+    fn fmt(&self, f: &mut Formatter<'_>) -> std::fmt::Result {
+        Display::fmt(self, f)
+    }
+}
+
+impl<F: OEF<D>, const D: usize> Neg for ExtensionAlgebra<F, D> {
+    type Output = Self;
+
+    #[inline]
+    fn neg(self) -> Self {
+        let mut arr = self.0;
+        arr.iter_mut().for_each(|x| *x = -*x);
+        Self(arr)
+    }
+}
+
+impl<F: OEF<D>, const D: usize> Add for ExtensionAlgebra<F, D> {
+    type Output = Self;
+
+    #[inline]
+    fn add(self, rhs: Self) -> Self {
+        let mut arr = self.0;
+        arr.iter_mut().zip(&rhs.0).for_each(|(x, &y)| *x += y);
+        Self(arr)
+    }
+}
+
+impl<F: OEF<D>, const D: usize> AddAssign for ExtensionAlgebra<F, D> {
+    fn add_assign(&mut self, rhs: Self) {
+        *self = *self + rhs;
+    }
+}
+
+impl<F: OEF<D>, const D: usize> Sum for ExtensionAlgebra<F, D> {
+    fn sum<I: Iterator<Item = Self>>(iter: I) -> Self {
+        iter.fold(Self::ZERO, |acc, x| acc + x)
+    }
+}
+
+impl<F: OEF<D>, const D: usize> Sub for ExtensionAlgebra<F, D> {
+    type Output = Self;
+
+    #[inline]
+    fn sub(self, rhs: Self) -> Self {
+        let mut arr = self.0;
+        arr.iter_mut().zip(&rhs.0).for_each(|(x, &y)| *x -= y);
+        Self(arr)
+    }
+}
+
+impl<F: OEF<D>, const D: usize> SubAssign for ExtensionAlgebra<F, D> {
+    #[inline]
+    fn sub_assign(&mut self, rhs: Self) {
+        *self = *self - rhs;
+    }
+}
+
+impl<F: OEF<D>, const D: usize> Mul for ExtensionAlgebra<F, D> {
+    type Output = Self;
+
+    #[inline]
+    fn mul(self, rhs: Self) -> Self {
+        let mut res = [F::ZERO; D];
+        let w = F::from_basefield(F::W);
+        for i in 0..D {
+            for j in 0..D {
+                res[(i + j) % D] += if i + j < D {
+                    self.0[i] * rhs.0[j]
+                } else {
+                    w * self.0[i] * rhs.0[i]
+                }
+            }
+        }
+        Self(res)
+    }
+}
+
+impl<F: OEF<D>, const D: usize> MulAssign for ExtensionAlgebra<F, D> {
+    #[inline]
+    fn mul_assign(&mut self, rhs: Self) {
+        *self = *self * rhs;
+    }
+}
+
+impl<F: OEF<D>, const D: usize> Product for ExtensionAlgebra<F, D> {
+    fn product<I: Iterator<Item = Self>>(iter: I) -> Self {
+        iter.fold(Self::one(), |acc, x| acc * x)
+    }
+}
+
+/// A polynomial in coefficient form.
+#[derive(Clone, Debug)]
+pub struct PolynomialCoeffsAlgebra<F: OEF<D>, const D: usize> {
+    pub(crate) coeffs: Vec<ExtensionAlgebra<F, D>>,
+}
+
+impl<F: OEF<D>, const D: usize> PolynomialCoeffsAlgebra<F, D> {
+    pub fn new(coeffs: Vec<ExtensionAlgebra<F, D>>) -> Self {
+        PolynomialCoeffsAlgebra { coeffs }
+    }
+
+    pub fn eval(&self, x: ExtensionAlgebra<F, D>) -> ExtensionAlgebra<F, D> {
+        self.coeffs
+            .iter()
+            .rev()
+            .fold(ExtensionAlgebra::ZERO, |acc, &c| acc * x + c)
+    }
+}
diff --git a/src/field/extension_field/mod.rs b/src/field/extension_field/mod.rs
index 7e74be78..c5510ea1 100644
--- a/src/field/extension_field/mod.rs
+++ b/src/field/extension_field/mod.rs
@@ -1,5 +1,6 @@
 use crate::field::field::Field;
 
+pub mod algebra;
 pub mod quadratic;
 pub mod quartic;
 mod quartic_quartic;
diff --git a/src/gates/interpolation.rs b/src/gates/interpolation.rs
index 53e0b7e7..3872b9ee 100644
--- a/src/gates/interpolation.rs
+++ b/src/gates/interpolation.rs
@@ -3,6 +3,7 @@ use std::marker::PhantomData;
 use std::ops::Range;
 
 use crate::circuit_builder::CircuitBuilder;
+use crate::field::extension_field::algebra::PolynomialCoeffsAlgebra;
 use crate::field::extension_field::target::ExtensionTarget;
 use crate::field::extension_field::{Extendable, FieldExtension};
 use crate::field::lagrange::interpolant;
@@ -111,19 +112,19 @@ where
         let mut constraints = Vec::with_capacity(self.num_constraints());
 
         let coeffs = (0..self.num_points)
-            .map(|i| vars.get_local_ext_ext(self.wires_coeff(i)))
+            .map(|i| vars.get_local_ext_algebra(self.wires_coeff(i)))
             .collect();
-        let interpolant = PolynomialCoeffs::new(coeffs);
+        let interpolant = PolynomialCoeffsAlgebra::new(coeffs);
 
         for i in 0..self.num_points {
             let point = vars.local_wires[self.wire_point(i)];
-            let value = vars.get_local_ext_ext(self.wires_value(i));
+            let value = vars.get_local_ext_algebra(self.wires_value(i));
             let computed_value = interpolant.eval(point.into());
             constraints.extend(&(value - computed_value).to_basefield_array());
         }
 
-        let evaluation_point = vars.get_local_ext_ext(self.wires_evaluation_point());
-        let evaluation_value = vars.get_local_ext_ext(self.wires_evaluation_value());
+        let evaluation_point = vars.get_local_ext_algebra(self.wires_evaluation_point());
+        let evaluation_value = vars.get_local_ext_algebra(self.wires_evaluation_value());
         let computed_evaluation_value = interpolant.eval(evaluation_point);
         constraints.extend(&(evaluation_value - computed_evaluation_value).to_basefield_array());
 
diff --git a/src/vars.rs b/src/vars.rs
index 88d0759a..1ac7935c 100644
--- a/src/vars.rs
+++ b/src/vars.rs
@@ -1,6 +1,7 @@
 use std::convert::TryInto;
 use std::ops::Range;
 
+use crate::field::extension_field::algebra::ExtensionAlgebra;
 use crate::field::extension_field::target::{ExtensionExtensionTarget, ExtensionTarget};
 use crate::field::extension_field::{Extendable, FieldExtension};
 use crate::field::field::Field;
@@ -18,16 +19,13 @@ pub struct EvaluationVarsBase<'a, F: Field> {
 }
 
 impl<'a, F: Extendable<D>, const D: usize> EvaluationVars<'a, F, D> {
-    pub fn get_local_ext_ext(
+    pub fn get_local_ext_algebra(
         &self,
         wire_range: Range<usize>,
-    ) -> <<F as Extendable<D>>::Extension as Extendable<D>>::Extension
-    where
-        F::Extension: Extendable<D>,
-    {
+    ) -> ExtensionAlgebra<F::Extension, D> {
         debug_assert_eq!(wire_range.len(), D);
         let arr = self.local_wires[wire_range].try_into().unwrap();
-        <<F as Extendable<D>>::Extension as Extendable<D>>::Extension::from_basefield_array(arr)
+        ExtensionAlgebra::from_basefield_array(arr)
     }
 }