A 40-year-old lady who has been a smoker has a hemoglobin level of 19. What hormone should you check?

In this scenario, checking the level of erythropoietin is the most pertinent choice given the patient's elevated hemoglobin level of 19 g/dL. Erythropoietin is a hormone produced primarily by the kidneys in response to low oxygen levels in the blood, or hypoxia, and is responsible for stimulating the production of red blood cells in the bone marrow.

In smokers, elevated hemoglobin can often result from chronic hypoxia due to reduced oxygen availability caused by smoking, leading to increased erythropoietin stimulation and consequently higher red blood cell production. Evaluating erythropoietin levels can help determine whether the body's response to the hypoxic condition is appropriate or if there are other underlying factors contributing to the elevated hemoglobin.

The other hormones listed would not provide insights relevant to this particular clinical situation. While aldosterone is involved in fluid and electrolyte balance, cortisol is related to stress response and metabolism, and T4 (thyroxine) is a thyroid hormone that regulates metabolism, none of these are likely to directly correlate with an elevated hemoglobin level in the context of chronic smoking. Thus, measuring erythropoietin is essential in determining the cause of the elevated hemoglobin in this patient.