Advanced Solutions: Mastering Complex Math Problems
For master's level students, tackling theoretical mathematics questions is essential for mastering the subject. Here, we address two advanced theoretical questions in mathematics and provide detailed solutions. For additional assistance with complex problems, consider reaching out to mathsassignmenthelp.com, where expert "Math Assignment Solver" services are available to support your academic needs.

Proving Every Subgroup of an Abelian Group is Normal
Question:
Let G be an abelian group. Prove that every subgroup of G is normal.

Answer:
To prove that every subgroup of an abelian group G is normal, we need to understand the concept of a normal subgroup. A subgroup H of G is normal if, for every element g in G and every element h in H, the element obtained by conjugating h with g is still in H.

Since G is abelian, it means that the group operation is commutative. This implies that for any elements g and h in G, the order of the operation does not matter, so g multiplied by h is the same as h multiplied by g.

Now consider any subgroup H of G. To show that H is normal, we need to demonstrate that the conjugated element, which is obtained by multiplying h on the left and right by g, is still within H.

In an abelian group, because the operation is commutative, the conjugated element simplifies to just h itself. This is because the operation g multiplied by h is the same as h multiplied by g, and thus, conjugating h by g does not change h.

Therefore, since the conjugated element is simply h, it remains in H, proving that H is a normal subgroup of G.

Proving Every Non-Constant Polynomial with Complex Coefficients Has at Least One Complex Root
Question:
Prove that every non-constant polynomial with complex coefficients has at least one complex root.

Answer:
The Fundamental Theorem of Algebra asserts that every non-constant polynomial with complex coefficients has at least one complex root. To prove this, we rely on the completeness property of the complex numbers.

Consider a polynomial of degree n with complex coefficients. If this polynomial did not have any roots, then it would imply that the polynomial function never crosses the horizontal axis in the complex plane.

However, complex polynomials are continuous functions, and in the field of complex numbers, every polynomial function is also bounded, meaning that its values are confined within a specific range. If a polynomial were to avoid zero, then by continuity, it would imply that the polynomial function would need to be constant, which contradicts the assumption of it being non-constant.

Thus, by the completeness of the complex number system and the properties of continuous functions, we conclude that every non-constant polynomial must have at least one complex root. This root could be found by factoring the polynomial or using various algebraic techniques, but it always exists in the complex number system.
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