FirebaseUser not an existing class | authentication errors

Question

I am trying to authticate users with firebase, and after the user fills out the form fields, this is how I am sending the user to the next page, after registering the user with my firebase database.

onPressed: () async {
          try {
            FirebaseUser user =
                (await FirebaseAuth.instance.createUserWithEmailAndPassword(
              email: _emailController.text,
              password: _passwordController.text,
            ))
                    .user;
            if (user != null) {
              UserUpdateInfo updateUser = UserUpdateInfo();
              updateUser.displayName = _usernameController.text;
              user.updateProfile(updateUser);
              Navigator.of(context).pushNamed(AppRoutes.Homepage_1);
            }

According to docs, the approach above looks correct, but I am getting a number of errors, such as

Undefined class 'FirebaseUser'.
Try changing the name to the name of an existing class, or creating a class with the name 'FirebaseUser'.

Undefined class 'UserUpdateInfo'.
Try changing the name to the name of an existing class, or creating a class with the name 'UserUpdateInfo'.

The method 'UserUpdateInfo' isn't defined for the type '_SignUpViewState'.
Try correcting the name to the name of an existing method, or defining a method named 'UserUpdateInfo'.

These error messages are not making sense as to why they are being thrown, as I have followed the relevant docs, and others online, have used this same code with no problems.

Am I missing something simple, or is something missing entirely from my broader code?

class SignUp extends StatefulWidget {
  @override
  _SignUpViewState createState() => _SignUpViewState();
}

class _SignUpViewState extends State<SignUp> {
  final _formKey = GlobalKey<FormState>();
  TextEditingController _firstnameController = TextEditingController();
  TextEditingController _emailController = TextEditingController();
  TextEditingController _passwordController = TextEditingController();
  TextEditingController _repasswordController = TextEditingController();

  @override
  Widget build(BuildContext context) {
    final mq = MediaQuery.of(context);

    final logo = Image.asset(
      "assets/logo.png",
      height: mq.size.height / 4,
    );

    final firstnameField = TextFormField(
      controller: _firstnameController,
      style: TextStyle(
        color: Colors.white,
      ),
      cursorColor: Colors.white,
      decoration: InputDecoration(
        focusedBorder: UnderlineInputBorder(
          borderSide: BorderSide(
            color: Colors.white,
          ),
        ),
        labelText: "First Name",
        labelStyle: TextStyle(
          color: Colors.white,
        ),
        hintStyle: TextStyle(
          color: Colors.white,
        ),
      ),
    );

    final emailField = TextFormField(
      controller: _emailController,
      keyboardType: TextInputType.emailAddress,
      style: TextStyle(
        color: Colors.white,
      ),
      cursorColor: Colors.white,
      decoration: InputDecoration(
        focusedBorder: UnderlineInputBorder(
          borderSide: BorderSide(
            color: Colors.white,
          ),
        ),
        labelText: "Email",
        labelStyle: TextStyle(
          color: Colors.white,
        ),
        hintStyle: TextStyle(
          color: Colors.white,
        ),
      ),
    );

    final passwordField = TextFormField(
      obscureText: true,
      controller: _passwordController,
      style: TextStyle(
        color: Colors.white,
      ),
      cursorColor: Colors.white,
      decoration: InputDecoration(
        focusedBorder: UnderlineInputBorder(
          borderSide: BorderSide(
            color: Colors.white,
          ),
        ),
        labelText: "Password",
        labelStyle: TextStyle(
          color: Colors.white,
        ),
        hintStyle: TextStyle(
          color: Colors.white,
        ),
      ),
    );

    final repasswordField = TextFormField(
      obscureText: true,
      controller: _repasswordController,
      style: TextStyle(
        color: Colors.white,
      ),
      cursorColor: Colors.white,
      decoration: InputDecoration(
        focusedBorder: UnderlineInputBorder(
          borderSide: BorderSide(
            color: Colors.white,
          ),
        ),
        hintText: "password",
        labelStyle: TextStyle(
          color: Colors.white,
        ),
        hintStyle: TextStyle(
          color: Colors.white,
        ),
      ),
    );

    final fields = Padding(
      padding: EdgeInsets.only(top: 10.0),
      child: Column(
        mainAxisAlignment: MainAxisAlignment.start,
        children: <Widget>[
          firstnameField,
          emailField,
          passwordField,
          repasswordField,
        ],
      ),
    );

    final registerButton = Material(
      elevation: 5.0,
      borderRadius: BorderRadius.circular(25.0),
      color: Colors.white,
      child: MaterialButton(
        minWidth: mq.size.width / 1.2,
        padding: EdgeInsets.fromLTRB(10.0, 15.0, 10.0, 15.0),
        child: Text(
          "Register",
          textAlign: TextAlign.center,
          style: TextStyle(
            fontSize: 20.0,
            color: Colors.black,
            fontWeight: FontWeight.bold,
          ),
        ),
        onPressed: () async {
          try {
            FirebaseUser user =
                (await FirebaseAuth.instance.createUserWithEmailAndPassword(
              email: _emailController.text,
              password: _passwordController.text,
            ))
                    .user;
            if (user != null) {
              UserUpdateInfo updateUser = UserUpdateInfo();
              updateUser.displayName = _firstnameController.text;
              user.updateProfile(updateUser);
              Navigator.of(context).pushNamed(AppRoutes.Homepage_1);
            }
          } catch (e) {
            print(e);
            _firstnameController.text =
                ""; //these reset the responses when there is an error in the form;LOOK HERE
            _passwordController.text = "";
            _repasswordController.text = "";
            _emailController.text = "";
            // TODO: alertdialog with error
          }
        },
      ),
    );

    final bottom = Column(
      mainAxisAlignment: MainAxisAlignment.start,
      crossAxisAlignment: CrossAxisAlignment.stretch,
      children: <Widget>[
        registerButton,
        Padding(
          padding: EdgeInsets.all(8.0),
        ),
        Row(
          mainAxisAlignment: MainAxisAlignment.spaceBetween,
          children: <Widget>[
            Text(
              "Already have an account?",
              style: Theme.of(context).textTheme.subtitle1.copyWith(
                    color: Colors.white,
                  ),
            ),
            MaterialButton(
              onPressed: () {
                Navigator.of(context).pushNamed(AppRoutes.authLogin);
              },
              child: Text(
                "Login",
                style: Theme.of(context).textTheme.subtitle1.copyWith(
                      color: Colors.white,
                      decoration: TextDecoration.underline,
                    ),
              ),
            ),
          ],
        ),
      ],
    );

    return Scaffold(
      backgroundColor: Color(0xff8c52ff),
      body: Form(
        key: _formKey,
        child: SingleChildScrollView(
          padding: EdgeInsets.all(36),
          child: Container(
            height: mq.size.height,
            child: Column(
              mainAxisAlignment: MainAxisAlignment.spaceAround,
              children: <Widget>[
                logo,
                fields,
                Padding(
                  padding: EdgeInsets.only(bottom: 150),
                  child: bottom,
                ),
              ],
            ),
          ),
        ),
      ),
    );
  }
}