diff --git a/src/bls12_381/fq2.rs b/src/bls12_381/fq2.rs
index 3095661..aa6ccde 100644
--- a/src/bls12_381/fq2.rs
+++ b/src/bls12_381/fq2.rs
@@ -158,7 +158,7 @@ impl Field for Fq2 {
 impl SqrtField for Fq2 {
 
     fn legendre(&self) -> ::LegendreSymbol {
-        Fq2::norm(&self).legendre()
+        self.norm().legendre()
     }
 
     fn sqrt(&self) -> Option<Self> {
@@ -578,7 +578,7 @@ fn bench_fq2_sqrt(b: &mut ::test::Bencher) {
 #[test]
 fn fq2_field_tests() {
     use ::PrimeField;
-    
+
     ::tests::field::random_field_tests::<Fq2>();
     ::tests::field::random_sqrt_tests::<Fq2>();
     ::tests::field::random_frobenius_tests::<Fq2, _>(super::fq::Fq::char(), 13);
diff --git a/src/bls12_381/fr.rs b/src/bls12_381/fr.rs
index 0f159de..058cb6a 100644
--- a/src/bls12_381/fr.rs
+++ b/src/bls12_381/fr.rs
@@ -554,6 +554,7 @@ impl Fr {
 impl SqrtField for Fr {
 
     fn legendre(&self) -> ::LegendreSymbol {
+        // s = self^((r - 1) // 2)
         let s = self.pow([0x7fffffff80000000, 0xa9ded2017fff2dff, 0x199cec0404d0ec02, 0x39f6d3a994cebea4]);
         if s == Self::zero() { Zero }
         else if s == Self::one() { QuadraticResidue }
diff --git a/src/tests/field.rs b/src/tests/field.rs
index 5f99992..bddb93e 100644
--- a/src/tests/field.rs
+++ b/src/tests/field.rs
@@ -1,5 +1,5 @@
 use rand::{Rng, SeedableRng, XorShiftRng};
-use ::{SqrtField, Field, PrimeField};
+use ::{SqrtField, Field, PrimeField, LegendreSymbol};
 
 pub fn random_frobenius_tests<F: Field, C: AsRef<[u64]>>(characteristic: C, maxpower: usize) {
     let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
@@ -26,6 +26,7 @@ pub fn random_sqrt_tests<F: SqrtField>() {
         let a = F::rand(&mut rng);
         let mut b = a;
         b.square();
+        assert_eq!(b.legendre(), LegendreSymbol::QuadraticResidue);
 
         let b = b.sqrt().unwrap();
         let mut negb = b;
@@ -38,6 +39,8 @@ pub fn random_sqrt_tests<F: SqrtField>() {
     for _ in 0..10000 {
         let mut b = c;
         b.square();
+        assert_eq!(b.legendre(), LegendreSymbol::QuadraticResidue);
+
         b = b.sqrt().unwrap();
 
         if b != c {