**Core Concept**
The treatment of congenital adrenal hyperplasia (CAH) due to 21 β-hydroxylase deficiency involves replacing the deficient enzyme's activity, thereby reducing androgen production. The goal is to decrease excessive androgen levels, which can lead to virilization in females, and prevent adrenal crisis.

**Why the Correct Answer is Right**
Administering synthetic glucocorticoids, such as dexamethasone or prednisone, suppresses the hypothalamic-pituitary-adrenal (HPA) axis, reducing the secretion of adrenocorticotropic hormone (ACTH). This decrease in ACTH levels subsequently reduces the production of cortisol precursors, including 17-hydroxyprogesterone (17-OHP) and androstenedione, which are overproduced in 21 β-hydroxylase deficiency. By decreasing the substrate for the deficient enzyme, glucocorticoid therapy effectively reduces androgen production and prevents the progression of CAH.

**Why Each Wrong Option is Incorrect**
**Option A:** Incorrect because glucocorticoids do not directly increase 21 β-hydroxylase activity. While glucocorticoids have a negative feedback effect on the HPA axis, they do not correct the enzymatic deficiency.

**Option B:** Incorrect because mineralocorticoids, such as fludrocortisone, are not the primary treatment for CAH due to 21 β-hydroxylase deficiency. Mineralocorticoids are used to replace aldosterone activity in patients with aldosterone deficiency, but they do not address androgen overproduction.

**Option C:** Incorrect because glucocorticoids do not directly suppress adrenal androgen production. While glucocorticoids reduce ACTH levels and subsequently decrease adrenal androgen production, this is an indirect effect rather than a direct action on the adrenal gland.

**Option D:** Incorrect because dexamethasone is a synthetic glucocorticoid, not a mineralocorticoid. While dexamethasone has mineralocorticoid activity at high doses, it is primarily used as a glucocorticoid in the treatment of CAH.

**Clinical Pearl / High-Yield Fact**
It is essential to remember that glucocorticoid therapy in CAH due to 21 β-hydroxylase deficiency is a long-term treatment that requires careful monitoring of growth, development, and adrenal function. Regular follow-up and adjustments to glucocorticoid doses are necessary to prevent complications and ensure optimal growth and development.

**Correct Answer:** C.