diff --git a/src/field/extension_field/algebra.rs b/src/field/extension_field/algebra.rs
index 004e8dc1..97c3c12d 100644
--- a/src/field/extension_field/algebra.rs
+++ b/src/field/extension_field/algebra.rs
@@ -151,3 +151,101 @@ impl<F: OEF<D>, const D: usize> PolynomialCoeffsAlgebra<F, D> {
             .fold(ExtensionAlgebra::ZERO, |acc, &c| acc * x + c)
     }
 }
+
+#[cfg(test)]
+mod tests {
+    use crate::field::crandall_field::CrandallField;
+    use crate::field::extension_field::algebra::ExtensionAlgebra;
+    use crate::field::extension_field::{Extendable, FieldExtension};
+    use crate::field::field::Field;
+    use itertools::Itertools;
+
+    /// Tests that the multiplication on the extension algebra lifts that of the field extension.
+    fn test_extension_algebra<F: Extendable<D>, const D: usize>() {
+        #[derive(Copy, Clone, Debug)]
+        enum ZeroOne {
+            Zero,
+            One,
+        }
+
+        let to_field = |zo: &ZeroOne| match zo {
+            ZeroOne::Zero => F::ZERO,
+            ZeroOne::One => F::ONE,
+        };
+        let to_fields = |x: &[ZeroOne], y: &[ZeroOne]| -> (F::Extension, F::Extension) {
+            let mut arr0 = [F::ZERO; D];
+            let mut arr1 = [F::ZERO; D];
+            arr0.copy_from_slice(&x.iter().map(to_field).collect::<Vec<_>>());
+            arr1.copy_from_slice(&y.iter().map(to_field).collect::<Vec<_>>());
+            (
+                <F as Extendable<D>>::Extension::from_basefield_array(arr0),
+                <F as Extendable<D>>::Extension::from_basefield_array(arr1),
+            )
+        };
+
+        // Standard MLE formula.
+        let selector = |xs: Vec<ZeroOne>, ts: &[F::Extension]| -> F::Extension {
+            (0..2 * D)
+                .map(|i| match xs[i] {
+                    ZeroOne::Zero => F::Extension::ONE - ts[i],
+                    ZeroOne::One => ts[i],
+                })
+                .product()
+        };
+
+        let mul_mle = |ts: Vec<F::Extension>| -> [F::Extension; D] {
+            let mut ans = [F::Extension::ZERO; D];
+            for xs in (0..2 * D)
+                .map(|_| vec![ZeroOne::Zero, ZeroOne::One])
+                .multi_cartesian_product()
+            {
+                let (a, b) = to_fields(&xs[..D], &xs[D..]);
+                let c = a * b;
+                let res = selector(xs, &ts);
+                for i in 0..D {
+                    ans[i] += res * c.to_basefield_array()[i].into();
+                }
+            }
+            ans
+        };
+
+        let ts = F::Extension::rand_vec(2 * D);
+        let mut arr0 = [F::Extension::ZERO; D];
+        let mut arr1 = [F::Extension::ZERO; D];
+        arr0.copy_from_slice(&ts[..D]);
+        arr1.copy_from_slice(&ts[D..]);
+        let x = ExtensionAlgebra::from_basefield_array(arr0);
+        let y = ExtensionAlgebra::from_basefield_array(arr1);
+        let z = x * y;
+
+        dbg!(z.0, mul_mle(ts.clone()));
+        assert_eq!(z.0, mul_mle(ts));
+    }
+
+    mod base {
+        use super::*;
+
+        #[test]
+        fn test_algebra() {
+            test_extension_algebra::<CrandallField, 1>();
+        }
+    }
+
+    mod quadratic {
+        use super::*;
+
+        #[test]
+        fn test_algebra() {
+            test_extension_algebra::<CrandallField, 2>();
+        }
+    }
+
+    mod quartic {
+        use super::*;
+
+        #[test]
+        fn test_algebra() {
+            test_extension_algebra::<CrandallField, 4>();
+        }
+    }
+}