move conj_sqrt() to method for finite fields

src/sage/combinat/symmetric_group_algebra.py

@@ -1301,7 +1301,7 @@
         and also projective modules)::
 
             sage: SGA = SymmetricGroupAlgebra(QQ, 5)
             sage: for la in Partitions(SGA.n):
             ....:     idem = SGA.ladder_idemponent(la)
             ....:     assert idem^2 == idem
             ....:     print(la, SGA.principal_ideal(idem).dimension())
@@ -2163,7 +2163,6 @@
             ...
             NotImplementedError: not implemented when p|n!; dimension of invariant forms may be greater than one
         """
-        from sage.matrix.special import diagonal_matrix
         F = self.base_ring()
         G = self.group()
 
@@ -2189,22 +2188,12 @@
             raise ValueError("the base ring must be a finite field of square order")
         if F.characteristic().divides(G.cardinality()):
             raise NotImplementedError("not implemented when p|n!; dimension of invariant forms may be greater than one")
-        q = F.order().sqrt()
-
-        def conj_square_root(u):
-            if not u:
-                return F.zero()
-            z = F.multiplicative_generator()
-            k = u.log(z)
-            if k % (q+1) != 0:
-                raise ValueError(f"unable to factor as {u} is not in base field GF({q})")
-            return z ** ((k//(q+1)) % (q-1))
 
         dft_matrix = self.dft()
         n = dft_matrix.nrows()
         for i in range(n):
             d = sum(dft_matrix[i, j] * dft_matrix[i, j].conjugate() for j in range(n))
-            dft_matrix[i] *= ~conj_square_root(d)
+            dft_matrix[i] *= ~d.conj_sqrt()
         return dft_matrix
 
     def _dft_seminormal(self, mult='l2r'):

src/sage/rings/finite_rings/element_givaro.pyx

@@ -1070,6 +1070,64 @@ cdef class FiniteField_givaroElement(FinitePolyExtElement):
         else:
             raise ValueError("must be a perfect square.")
 
+    def conj_sqrt(FiniteField_givaroElement self):
+        r"""
+        Return a conjugate square root of this finite field element in its
+        parent, if there is one.  Otherwise, raise a :exc:`ValueError`.
+
+        ALGORITHM:
+
+        ``self`` is stored as `a^k` for some generator `a`.
+        Return `a^{k/(q+1)}` for `k` divisible by `q+1`.
+
+        .. WARNING::
+
+            This is only implemented for elements whose exponent
+            is divisible by `q+1` in fields of order `q**2`.
+
+        EXAMPLES::
+
+            sage: k.<a> = GF(7**2)
+            sage: k(0).conj_sqrt()
+            0
+            sage: z = k(2).conj_sqrt(); z
+            a + 4
+            sage: z*z.conjugate()
+            2
+            sage: z = k(3).conj_sqrt(); z
+            a
+            sage: z*z.conjugate()
+            3
+            sage: z = k(4).conj_sqrt(); z
+            2*a + 6
+            sage: z*z.conjugate()
+            4
+
+        TESTS::
+
+            sage: k.<a> = GF(7**3)
+            sage: k(3).conj_sqrt()
+            Traceback (most recent call last):
+            ...
+            ValueError: the base ring must be a finite field of square order
+            sage: k.<a> = GF(7**2)
+            sage: a.conj_sqrt()
+            Traceback (most recent call last):
+            ...
+            ValueError: element must be element of base field GF(7)
+        """
+        if not self.parent().order().is_square():
+            raise ValueError("the base ring must be a finite field of square order")
+        q = self.parent().order().sqrt()
+
+        if self == 0:
+            return 0
+        z = self.parent().multiplicative_generator()
+        k = self.log(z)  # Compute discrete log of u to the base z
+        if k % (q+1) != 0:
+            raise ValueError(f"element must be element of base field GF({q})")
+        return z ** (k//(q+1))
+
     cpdef _add_(self, right):
         """
         Add two elements.