diff --git a/field/quadratic_extension.go b/field/quadratic_extension.go
new file mode 100644
index 0000000..46668e0
--- /dev/null
+++ b/field/quadratic_extension.go
@@ -0,0 +1,163 @@
+package field
+
+import (
+	"fmt"
+	"math/bits"
+
+	"github.com/consensys/gnark/frontend"
+)
+
+type QuadraticExtensionAPI struct {
+	fieldAPI frontend.API
+
+	W        F
+	DTH_ROOT F
+
+	ONE_QE  QuadraticExtension
+	ZERO_QE QuadraticExtension
+}
+
+func NewQuadraticExtensionAPI(fieldAPI frontend.API, degreeBits uint64) *QuadraticExtensionAPI {
+	// TODO:  Should degreeBits be verified that it fits within the field and that degree is within uint64?
+
+	return &QuadraticExtensionAPI{
+		fieldAPI: fieldAPI,
+
+		W:        NewFieldElement(7),
+		DTH_ROOT: NewFieldElement(18446744069414584320),
+
+		ONE_QE:  QuadraticExtension{ONE_F, ZERO_F},
+		ZERO_QE: QuadraticExtension{ZERO_F, ZERO_F},
+	}
+}
+
+func (c *QuadraticExtensionAPI) SquareExtension(a QuadraticExtension) QuadraticExtension {
+	return c.MulExtension(a, a)
+}
+
+func (c *QuadraticExtensionAPI) MulExtension(a QuadraticExtension, b QuadraticExtension) QuadraticExtension {
+	c_0 := c.fieldAPI.Add(c.fieldAPI.Mul(a[0], b[0]).(F), c.fieldAPI.Mul(c.W, a[1], b[1])).(F)
+	c_1 := c.fieldAPI.Add(c.fieldAPI.Mul(a[0], b[1]).(F), c.fieldAPI.Mul(a[1], b[0])).(F)
+	return QuadraticExtension{c_0, c_1}
+}
+
+func (c *QuadraticExtensionAPI) AddExtension(a QuadraticExtension, b QuadraticExtension) QuadraticExtension {
+	c_0 := c.fieldAPI.Add(a[0], b[0]).(F)
+	c_1 := c.fieldAPI.Add(a[1], b[1]).(F)
+	return QuadraticExtension{c_0, c_1}
+}
+
+func (c *QuadraticExtensionAPI) SubExtension(a QuadraticExtension, b QuadraticExtension) QuadraticExtension {
+	c_0 := c.fieldAPI.Sub(a[0], b[0]).(F)
+	c_1 := c.fieldAPI.Sub(a[1], b[1]).(F)
+	return QuadraticExtension{c_0, c_1}
+}
+
+func (c *QuadraticExtensionAPI) DivExtension(a QuadraticExtension, b QuadraticExtension) QuadraticExtension {
+	return c.MulExtension(a, c.InverseExtension(b))
+}
+
+func (c *QuadraticExtensionAPI) IsZero(a QuadraticExtension) frontend.Variable {
+	return c.fieldAPI.Mul(c.fieldAPI.IsZero(a[0]), c.fieldAPI.IsZero(a[1]))
+}
+
+// TODO: Instead of calculating the inverse within the circuit, can witness the
+// inverse and assert that a_inverse * a = 1.  Should reduce # of constraints.
+func (c *QuadraticExtensionAPI) InverseExtension(a QuadraticExtension) QuadraticExtension {
+	// First assert that a doesn't have 0 value coefficients
+	a0_is_zero := c.fieldAPI.IsZero(a[0])
+	a1_is_zero := c.fieldAPI.IsZero(a[1])
+
+	// assert that a0_is_zero OR a1_is_zero == false
+	c.fieldAPI.AssertIsEqual(c.fieldAPI.Mul(a0_is_zero, a1_is_zero).(F), ZERO_F)
+
+	a_pow_r_minus_1 := QuadraticExtension{a[0], c.fieldAPI.Mul(a[1], c.DTH_ROOT).(F)}
+	a_pow_r := c.MulExtension(a_pow_r_minus_1, a)
+	return c.ScalarMulExtension(a_pow_r_minus_1, c.fieldAPI.Inverse(a_pow_r[0]).(F))
+}
+
+func (c *QuadraticExtensionAPI) ScalarMulExtension(a QuadraticExtension, scalar F) QuadraticExtension {
+	return QuadraticExtension{c.fieldAPI.Mul(a[0], scalar).(F), c.fieldAPI.Mul(a[1], scalar).(F)}
+}
+
+func (c *QuadraticExtensionAPI) FieldToQE(a F) QuadraticExtension {
+	return QuadraticExtension{a, ZERO_F}
+}
+
+// / Exponentiate `base` to the power of a known `exponent`.
+func (c *QuadraticExtensionAPI) ExpU64Extension(a QuadraticExtension, exponent uint64) QuadraticExtension {
+	switch exponent {
+	case 0:
+		return c.ONE_QE
+	case 1:
+		return a
+	case 2:
+		return c.SquareExtension(a)
+	default:
+	}
+
+	current := a
+	product := c.ONE_QE
+
+	for i := 0; i < bits.Len64(exponent); i++ {
+		if i != 0 {
+			current = c.SquareExtension(current)
+		}
+
+		if (exponent >> i & 1) != 0 {
+			product = c.MulExtension(product, current)
+		}
+	}
+
+	return product
+}
+
+func (c *QuadraticExtensionAPI) ReduceWithPowers(terms []QuadraticExtension, scalar QuadraticExtension) QuadraticExtension {
+	sum := c.ZERO_QE
+
+	for i := len(terms) - 1; i >= 0; i-- {
+		sum = c.AddExtension(
+			c.MulExtension(
+				sum,
+				scalar,
+			),
+			terms[i],
+		)
+	}
+
+	return sum
+}
+
+func (c *QuadraticExtensionAPI) Select(b0 frontend.Variable, qe0, qe1 QuadraticExtension) QuadraticExtension {
+	var retQE QuadraticExtension
+
+	for i := 0; i < 2; i++ {
+		retQE[i] = c.fieldAPI.Select(b0, qe0[i], qe1[i]).(F)
+	}
+
+	return retQE
+}
+
+func (c *QuadraticExtensionAPI) Lookup2(b0 frontend.Variable, b1 frontend.Variable, qe0, qe1, qe2, qe3 QuadraticExtension) QuadraticExtension {
+	var retQE QuadraticExtension
+
+	for i := 0; i < 2; i++ {
+		retQE[i] = c.fieldAPI.Lookup2(b0, b1, qe0[i], qe1[i], qe2[i], qe3[i]).(F)
+	}
+
+	return retQE
+}
+
+func (c *QuadraticExtensionAPI) AssertIsEqual(a, b QuadraticExtension) {
+	for i := 0; i < 2; i++ {
+		c.fieldAPI.AssertIsEqual(a[0], b[0])
+	}
+}
+
+func (c *QuadraticExtensionAPI) Println(a QuadraticExtension) {
+	fmt.Print("Degree 0 coefficient")
+	c.fieldAPI.Println(a[0])
+
+	fmt.Print("Degree 1 coefficient")
+	c.fieldAPI.Println(a[1])
+}