You can indeed.
DISCLAIMER: Rooting may brick your device and invalidate your carrier and manufacturer warranty. I didn't write the guide below (full credit is due to the poster on XDA Developers, and I have left the words 'my' in place but it is HIS work), and it is your responsibility to check you are happy to proceed. HERE BE DRAGONS.
Rooting form a PC is easier and has less scope for human errors and is liklier to work in general.
This XDA thread is relating to the sprint version but should apply to the S3 general.
Prerequisites
The following files are attached to this post for convenience. Download and install the apps, download and extract the zip file somewhere convenient. Superuser by Koush(MUST be this one. You can change it later if you so desire.) Terminal Emulator by Jack Palevich ( MUST be this one) Zip file, extracted somewhere convenient
THE GOODS
These generic names will be used: /path/to/motochopperDir - This is the full path to the extracted motochopper-mobile DIRECTORY or FOLDER. Example: /sdcard/motochopper-mobile (will vary depending on where you extracted the files).
NOTE: After running the script, Motochopper may or may not print "Failed", but the exploit DOES indeed work regardless of this. Make sure you reboot for changes to take effect.
Download the attached zip file AND both apps. Install Terminal Emulator and Superuser (use a file manager, navigate to the downloaded .apk files and click on each file to install them). Open up Terminal Emulator and type the following commands. Press the ENTER key after each command.
Code:
cd /path/to/motochopperDir
cat setup.sh > /data/data/jackpal.androidterm/setup.sh
cd /data/data/jackpal.androidterm
chmod 755 setup.sh
./setup.sh /path/to/motochopperDir
From there, my setup.sh script will handle the rest of the process. If you are still unsure, see my sample run below.
Sample run (files were extracted to /sdcard/moto)
cd /sdcard/moto
cat setup.sh > /data/data/jackpal.androidterm/setup.sh
cd /data/data/jackpal.androidterm
chmod 755 setup.sh
./setup.sh /sdcard/moto
The Links: