diff --git a/ext/build.sh b/ext/build.sh
index 2c22393..b9994db 100644
--- a/ext/build.sh
+++ b/ext/build.sh
@@ -5,7 +5,7 @@ set -euo pipefail
 echo "=== Building C libraries ==="
 # Adapted from https://github.com/fredrik-johansson/python-flint/blob/00699afa47aaa4c56e42cb98a1f8231e9000eddd/bin/build_dependencies_unix.sh
 
-PREFIX=$(pwd)/tmlt/core/ext
+PREFIX=$(pwd)/src/tmlt/core/ext
 echo $PREFIX
 mkdir -p $PREFIX/src
 source ext/dependency_versions.sh
@@ -18,68 +18,49 @@ case "$(uname)" in
     *) nproc=$(getconf _NPROCESSORS_ONLN) || nproc=2
 esac
 
-if [[ "$(uname)" = "Darwin" ]]; then
-    FLINTARB_WITHGMP="--with-gmp=$PREFIX"
-
-    # GMP
-    if [[ ! -f $PREFIX/lib/GMPVER || "$(cat $PREFIX/lib/GMPVER)" != "$GMPVER" ]]; then
-        curl -O https://gmplib.org/download/gmp/gmp-$GMPVER.tar.xz
-        tar xf gmp-$GMPVER.tar.xz
-        pushd gmp-$GMPVER
-        # Show the output of configfsf.guess
-        ./configfsf.guess
-        ./configure --prefix=$PREFIX --enable-fat --enable-shared=yes --enable-static=no --host=x86_64-apple-darwin
-        make -j $nproc
-        make install
-        echo "$GMPVER" > $PREFIX/lib/GMPVER
-        rm -f $PREFIX/lib/MPFRVER
-        popd
-    else
-        echo "Using existing GMP..."
-    fi
-else
-    if ! (command -v make && command -v curl && command -v bzip2 && command -v m4) >/dev/null
-    then
-        echo "make, curl, bzip2, and m4 are required to build dependencies from source."
-        exit 1
-    fi
-
-    FLINTARB_WITHGMP="--with-mpir=$PREFIX"
+if ! (command -v make && command -v curl && command -v m4) >/dev/null
+then
+    echo "make, curl, and m4 are required to build dependencies from source."
+    exit 1
+fi
 
-    # YASM (dependency of MPIR)
-    if [[ ! -f $PREFIX/lib/YASMVER || "$(cat $PREFIX/lib/YASMVER)" != "$YASMVER" ]]; then
-        curl -L -O https://github.com/yasm/yasm/releases/download/v$YASMVER/yasm-$YASMVER.tar.gz
-        tar xf yasm-$YASMVER.tar.gz
-        pushd yasm-$YASMVER
-        ./configure --prefix=$PREFIX
-        make -j $nproc
-        make install
-        echo "$YASMVER" > $PREFIX/lib/YASMVER
-        rm -f $PREFIX/lib/MPIRVER
-        popd
-    else
-        echo "Using existing YASM..."
+# On Linux, cibuildwheel compiles the dependencies and builds platform-specific wheels on a Docker
+# container. Since the dependencies built when running `poetry install` are not compatible with the
+# Docker container, we need to rebuild them here. We use AUDITWHEEL_PLAT to determine whether the
+# dependencies need to be rebuilt.
+AUDITWHEEL_PLAT=${AUDITWHEEL_PLAT-}
+if [[ "$(uname)" = "Linux" ]]; then
+    if [[ ! -f $PREFIX/lib/AUDITWHEEL_PLAT || "$(cat $PREFIX/lib/AUDITWHEEL_PLAT)" != "$AUDITWHEEL_PLAT" ]]; then
+        echo "Rebuilding dependencies because $PREFIX/lib/AUDITWHEEL_PLAT does not match $AUDITWHEEL_PLAT."
+        rm -f $PREFIX/lib/GMPVER $PREFIX/lib/MPFRVER $PREFIX/lib/FLINTVER $PREFIX/lib/ARBVER
     fi
+fi
 
-    # MPIR
-    if [[ ! -f $PREFIX/lib/MPIRVER || "$(cat $PREFIX/lib/MPIRVER)" != "$MPIRVER" ]]; then
-        curl -O https://mpir.org/mpir-$MPIRVER.tar.bz2
-        tar xf mpir-$MPIRVER.tar.bz2
-        pushd mpir-$MPIRVER
-        ./configure --prefix=$PREFIX --with-yasm=$PREFIX/bin/yasm --enable-fat --enable-shared=yes --enable-static=no --enable-gmpcompat
-        make -j $nproc
-        make install
-        echo "$MPIRVER" > $PREFIX/lib/MPIRVER
-        rm -f $PREFIX/lib/MPFRVER
-        popd
-    else
-        echo "Using existing MPIR..."
+# GMP
+if [[ ! -f $PREFIX/lib/GMPVER || "$(cat $PREFIX/lib/GMPVER)" != "$GMPVER" ]]; then
+    configure_args="--prefix=$PREFIX --enable-fat --enable-shared=yes --enable-static=no"
+    if [[ "$(uname)" = "Darwin" ]]; then
+        configure_args="$configure_args --host=x86_64-apple-darwin"
     fi
+    curl -OLf https://gmplib.org/download/gmp/gmp-$GMPVER.tar.xz
+    tar xf gmp-$GMPVER.tar.xz
+    pushd gmp-$GMPVER
+    # Show the output of configfsf.guess
+    ./configfsf.guess
+    ./configure $configure_args
+    make -j $nproc
+    make check
+    make install
+    echo "$GMPVER" > $PREFIX/lib/GMPVER
+    rm -f $PREFIX/lib/MPFRVER
+    popd
+else
+    echo "Using existing GMP..."
 fi
 
 # MPFR
 if [[ ! -f $PREFIX/lib/MPFRVER || "$(cat $PREFIX/lib/MPFRVER)" != "$MPFRVER" ]]; then
-    curl -O https://ftp.gnu.org/gnu/mpfr/mpfr-$MPFRVER.tar.gz
+    curl -OLf https://ftp.gnu.org/gnu/mpfr/mpfr-$MPFRVER.tar.gz
     tar xf mpfr-$MPFRVER.tar.gz
     pushd mpfr-$MPFRVER
     ./configure --prefix=$PREFIX --with-gmp=$PREFIX --enable-shared=yes --enable-static=no
@@ -94,10 +75,10 @@ fi
 
 # FLINT
 if [[ ! -f $PREFIX/lib/FLINTVER || "$(cat $PREFIX/lib/FLINTVER)" != "$FLINTVER" ]]; then
-    curl -O https://www.flintlib.org/flint-$FLINTVER.tar.gz
+    curl -OLf https://www.flintlib.org/flint-$FLINTVER.tar.gz
     tar xf flint-$FLINTVER.tar.gz
     pushd flint-$FLINTVER
-    ./configure --prefix=$PREFIX $FLINTARB_WITHGMP --with-mpfr=$PREFIX --disable-static
+    ./configure --prefix=$PREFIX --with-gmp=$PREFIX --with-mpfr=$PREFIX --disable-static
     make -j $nproc
     make install
     echo "$FLINTVER" > $PREFIX/lib/FLINTVER
@@ -109,11 +90,11 @@ fi
 
 # Arb
 if [[ ! -f $PREFIX/lib/ARBVER || "$(cat $PREFIX/lib/ARBVER)" != "$ARBVER" ]]; then
-    curl -O -L https://github.com/fredrik-johansson/arb/archive/refs/tags/$ARBVER.tar.gz
+    curl -OLf https://github.com/fredrik-johansson/arb/archive/refs/tags/$ARBVER.tar.gz
     mv $ARBVER.tar.gz arb-$ARBVER.tar.gz
     tar xf arb-$ARBVER.tar.gz
     pushd arb-$ARBVER
-    ./configure --prefix=$PREFIX --with-flint=$PREFIX $FLINTARB_WITHGMP --with-mpfr=$PREFIX --disable-static
+    ./configure --prefix=$PREFIX --with-flint=$PREFIX --with-gmp=$PREFIX --with-mpfr=$PREFIX --disable-static
     make -j $nproc
     make install
     echo "$ARBVER" > $PREFIX/lib/ARBVER
@@ -122,6 +103,8 @@ else
     echo "Using existing Arb..."
 fi
 
+echo "$AUDITWHEEL_PLAT" > $PREFIX/lib/AUDITWHEEL_PLAT
+
 # The source archives for some of these libraries include Python files, which
 # can be spuriously picked up by our linters when run locally. There's no reason
 # to keep the sources around, so just delete them to get around this problem.
diff --git a/ext/dependency_versions.sh b/ext/dependency_versions.sh
index 8dd2d10..157408a 100644
--- a/ext/dependency_versions.sh
+++ b/ext/dependency_versions.sh
@@ -3,8 +3,6 @@
 # This file lists the versions of C libraries required for tmlt.core.utils.arb.
 
 GMPVER=6.2.1
-YASMVER=1.3.0
-MPIRVER=3.0.0
 MPFRVER=4.1.0
 FLINTVER=2.9.0
 ARBVER=2.23.0