Yes, your income is subject to Illinois tax; however, you are allowed some significant deductions on your Federal return. Your job in Alaska was temporary, which means that you continued to be an Illinois resident. What this means is that your travel, and stay, in Alaska are deductible expenses. A thorough consideration of this is found in this publication (click on link): Pub 463.
Here is an example of some of the deductions you will be entitled to:
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Travel between Illinois and Alaska. Plane tickets would be fully deductilbe, and if you rented a vehicle in Alaska, your car rental is deductible along with your gasoline spent for work. (If you drove your own vehicle to Alaska, you may deduct 54 cents/mile driven).
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Lodging costs. You may use your actual lodging expenses in Alaska, or you may use a per diem rate. These can be substantial deductions. Please see this website for the rates: http://www.defensetravel.dod.mil/pdcgi/pd-rates/opdrates5.cgi (You will need to use the town you stayed in, and use the Jan. 1- Sept. 30 rate). Remember, those rates are per day. You are allowed to count that amount as a deduction for each night you were away in Alaska, and you are allowed to deduct the full amount.
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Meal and Incidentals. This is also on that same chart, and is in addition to the Lodging and Travel Expense. Unlike lodging, you will only be able to get 50% of the actual rate as a deduction (although you do enter the full amount, and your deduction is still a signifiicant number).
These expenses are entered in Form 2106. This form can be accessed by following the instructions in the following FAQ: https://ttlc.intuit.com/replies/4800418. Once on the Form, you will look for the Travel Expense Screen. You will need to make the above calculations manually, but they can be entered as shown in this screenshot below:
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