From 753fa25ee22a327bffcf65981c797d2c8f4169d1 Mon Sep 17 00:00:00 2001
From: Max Horn <max@quendi.de>
Date: Thu, 17 Oct 2024 15:17:23 +0200
Subject: [PATCH] Implement isreal(elm::QQAbFieldElem) (#4207)

Also turn QQAbAutomorphism into immutable struct
---
 src/Rings/AbelianClosure.jl  | 4 +++-
 test/Rings/AbelianClosure.jl | 4 ++++
 2 files changed, 7 insertions(+), 1 deletion(-)

diff --git a/src/Rings/AbelianClosure.jl b/src/Rings/AbelianClosure.jl
index 662cb496601f..dd60211a93b8 100644
--- a/src/Rings/AbelianClosure.jl
+++ b/src/Rings/AbelianClosure.jl
@@ -1044,7 +1044,7 @@ end
 # Then $l = k a n_0 + b n_1$ is coprime to $n$ and has the properties
 # $l \equiv 1 \pmod{n_0}$ and $l \equiv k \pmod{n_1}$.
 
-mutable struct QQAbAutomorphism
+struct QQAbAutomorphism
   exp::Int
 end
 
@@ -1084,6 +1084,8 @@ end
 
 Base.conj(elm::QQAbFieldElem) = elm^QQAbAutomorphism(-1)
 
+Base.isreal(elm::QQAbFieldElem) = conj(elm) == elm
+
 ###############################################################################
 #
 #   Elements in quadratic subfields of cyclotomic fields
diff --git a/test/Rings/AbelianClosure.jl b/test/Rings/AbelianClosure.jl
index cb47f6b42fd8..9fb875fada9f 100644
--- a/test/Rings/AbelianClosure.jl
+++ b/test/Rings/AbelianClosure.jl
@@ -276,6 +276,10 @@ end
     f = hom(K, K, 3)
     @test f(a) == a^3
     @test f(z(9)) == z(9)
+
+    @test isone(conj(a)*a)
+    @test !isreal(a)
+    @test isreal(a+conj(a))
   end
 
   @testset "Square roots" begin